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A triangle is a
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
with three edges and three vertices. It is one of the basic
shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ...
s in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, any three points, when non-
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
, determine a unique triangle and simultaneously, a unique
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
(i.e. a two-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted.


Types of triangle

The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
. The names used for modern classification are either a direct transliteration of Euclid's Greek or their Latin translations.


By lengths of sides

Ancient Greek mathematician
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
defined three types of triangle according to the lengths of their sides:
gr, τῶν δὲ τριπλεύρων σχημάτων ἰσόπλευρον μὲν τρίγωνόν ἐστι τὸ τὰς τρεῖς ἴσας ἔχον πλευράς, ἰσοσκελὲς δὲ τὸ τὰς δύο μόνας ἴσας ἔχον πλευράς, σκαληνὸν δὲ τὸ τὰς τρεῖς ἀνίσους ἔχον πλευράς, lit= Of trilateral figures, an ''isopleuron'' quilateraltriangle is that which has its three sides equal, an ''isosceles'' that which has two of its sides alone equal, and a ''scalene'' that which has its three sides unequal.
* An ''
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
'' ( gr, ἰσόπλευρον, translit=isópleuron, lit=equal sides) has three sides of the same length. An equilateral triangle is also a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
with all angles measuring 60°. * An ''
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
'' ( gr, ἰσοσκελὲς, translit=isoskelés, lit=equal legs) has two sides of equal length.Euclid defines isosceles triangles based on the number of equal sides, i.e. ''only two equal sides''. An alternative approach defines isosceles triangles based on shared properties, i.e. ''equilateral triangles are a special case of isosceles triangles''. wikt:Isosceles triangle An isosceles triangle also has two angles of the same measure, namely the angles opposite to the two sides of the same length. This fact is the content of the
isosceles triangle theorem In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the ''pons asinorum'' (, ), typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in Eu ...
, which was known by
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with ''at least'' two equal sides. The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the
tetrakis square tiling In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed b ...
, is isosceles. * A ''scalene triangle'' ( gr, σκαληνὸν, translit=skalinón, lit=unequal) has all its sides of different lengths. Equivalently, it has all angles of different measure. Triangle.Equilateral.svg, Equilateral Triangle Triangle.Isosceles.svg, Isosceles triangle Triangle.Scalene.svg, Scalene triangle
Hatch marks Hatch marks (also called hash marks or tick marks) are a form of mathematical notation. They are used in three ways as: * Unit and value marks — as on a ruler or number line * Congruence notation in geometry — as on a geometric figure * Graph ...
, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of "ticks", short line segments in the form of
tally marks Tally marks, also called hash marks, are a unary numeral system ( arguably). They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate ...
; two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is usually no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, and a scalene triangle has different patterns on all sides since no sides are equal. Similarly, patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles: an equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, and a scalene triangle has different patterns on all angles, since no angles are equal.


By internal angles

Triangles can also be classified according to their
internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
s, measured here in degrees. * A ''
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
'' (or ''right-angled triangle'') has one of its interior angles measuring 90° (a
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
). The side opposite to the right angle is the
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
, the longest side of the triangle. The other two sides are called the ''legs'' or ''catheti'' (singular: ''
cathetus In a right triangle, a cathetus (originally from the Greek word ; plural: catheti), commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a "side about the right angle". The side opposite ...
'') of the triangle. Right triangles obey the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: , where ''a'' and ''b'' are the lengths of the legs and ''c'' is the length of the hypotenuse.
Special right triangles A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...
are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where . The 3–4–5 triangle is also known as the Egyptian triangle. In this situation, 3, 4, and 5 are a
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
. The other one is an isosceles triangle that has 2 angles measuring 45 degrees (45–45–90 triangle). ** Triangles that do not have an angle measuring 90° are called
oblique triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
s. * A triangle with all interior angles measuring less than 90° is an ''
acute triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
'' or ''acute-angled triangle''. If ''c'' is the length of the longest side, then , where ''a'' and ''b'' are the lengths of the other sides. * A triangle with one interior angle measuring more than 90° is an ''
obtuse triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
'' or ''obtuse-angled triangle''. If ''c'' is the length of the longest side, then , where ''a'' and ''b'' are the lengths of the other sides. * A triangle with an interior angle of 180° (and
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
vertices) is ''
degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
''. A right degenerate triangle has collinear vertices, two of which are coincident. A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and therefore is equilateral.


Basic facts

Triangles are assumed to be two-
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al
plane figure A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie o ...
s, unless the context provides otherwise (see Non-planar triangles, below). In rigorous treatments, a triangle is therefore called a ''2-
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
'' (see also
Polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
). Elementary facts about triangles were presented by
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
, in books 1–4 of his '' Elements'', written around 300 BC. The sum of the measures of the interior angles of a triangle in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
is always 180 degrees. This fact is equivalent to Euclid's
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...
. This allows determination of the measure of the third angle of any triangle, given the measure of two angles. An ''
exterior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
'' of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the
exterior angle theorem The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute ge ...
. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees.The ''n'' external angles of any ''n''-sided
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
polygon add up to 360 degrees.


Similarity and congruence

Two triangles are said to be '' similar'', if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity. Some basic
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s about similar triangles are: *
If and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
one pair of internal angles of two triangles have the same measure as each other, and another pair also have the same measure as each other, the triangles are similar. * If and only if one pair of corresponding sides of two triangles are in the same proportion as are another pair of corresponding sides, and their included angles have the same measure, then the triangles are similar. (The ''included angle'' for any two sides of a polygon is the internal angle between those two sides.) * If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar.Again, in all cases "mirror images" are also similar. Two triangles that are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
have exactly the same size and shape:All pairs of congruent triangles are also similar; but not all pairs of similar triangles are congruent. all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.) Some individually
necessary and sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
s for a pair of triangles to be congruent are: * SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure. * ASA: Two interior angles and the included side in a triangle have the same measure and length, respectively, as those in the other triangle. (The ''included side'' for a pair of angles is the side that is common to them.) * SSS: Each side of a triangle has the same length as a corresponding side of the other triangle. * AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to as ''AAcorrS'' and then includes ASA above.) Some individually sufficient conditions are: * Hypotenuse-Leg (HL) Theorem: The hypotenuse and a leg in a right triangle have the same length as those in another right triangle. This is also called RHS (right-angle, hypotenuse, side). * Hypotenuse-Angle Theorem: The hypotenuse and an acute angle in one right triangle have the same length and measure, respectively, as those in the other right triangle. This is just a particular case of the AAS theorem. An important condition is: * Side-Side-Angle (or Angle-Side-Side) condition: If two sides and a corresponding non-included angle of a triangle have the same length and measure, respectively, as those in another triangle, then this is ''not'' sufficient to prove congruence; but if the angle given is opposite to the longer side of the two sides, then the triangles are congruent. The Hypotenuse-Leg Theorem is a particular case of this criterion. The Side-Side-Angle condition does not by itself guarantee that the triangles are congruent because one triangle could be obtuse-angled and the other acute-angled. Using right triangles and the concept of similarity, the
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s sine and cosine can be defined. These are functions of an
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
which are investigated in
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
.


Right triangles

A central theorem is the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, which states in any
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
, the square of the length of the
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length ''c'', and the legs have lengths ''a'' and ''b'', then the theorem states that :a^2 + b^2 = c^2. The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle has a right angle opposite side ''c''. Some other facts about right triangles: * The acute angles of a right triangle are
complementary A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
. :: a + b + 90^\circ = 180^\circ \Rightarrow a + b = 90^\circ \Rightarrow a = 90^\circ - b. * If the legs of a right triangle have the same length, then the angles opposite those legs have the same measure. Since these angles are complementary, it follows that each measures 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times . * In a right triangle with acute angles measuring 30 and 60 degrees, the hypotenuse is twice the length of the shorter side, and the longer side is equal to the length of the shorter side times : ::c = 2a\, ::b = a\times\sqrt. For all triangles, angles and sides are related by the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
and
law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and a ...
(also called the ''cosine rule'' and ''sine rule'').


Existence of a triangle


Condition on the sides

The
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. That sum can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. It is not possible for that sum to be less than the length of the third side. A triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality.


Conditions on the angles

Three given angles form a non-degenerate triangle (and indeed an infinitude of them) if and only if both of these conditions hold: (a) each of the angles is positive, and (b) the angles sum to 180°. If degenerate triangles are permitted, angles of 0° are permitted.


Trigonometric conditions

Three positive angles ''α'', ''β'', and ''γ'', each of them less than 180°, are the angles of a triangle
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
any one of the following conditions holds: :\tan\tan+\tan\tan+\tan\tan=1,Vardan Verdiyan & Daniel Campos Salas, "Simple trigonometric substitutions with broad results", ''Mathematical Reflections'' no 6, 2007. :\sin^2+\sin^2+\sin^2+2\sin\sin\sin=1, :\sin(2\alpha) + \sin(2\beta) + \sin(2\gamma) = 4\sin(\alpha)\sin(\beta)\sin(\gamma), :\cos^2\alpha+\cos^2\beta+\cos^2\gamma+2\cos(\alpha)\cos(\beta)\cos(\gamma)=1, :\tan(\alpha) + \tan(\beta) + \tan(\gamma) = \tan(\alpha)\tan(\beta)\tan(\gamma), the last equality applying only if none of the angles is 90° (so the tangent function's value is always finite).


Points, lines, and circles associated with a triangle

There are thousands of different constructions that find a special point associated with (and often inside) a triangle, satisfying some unique property: see the article
Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the l ...
for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is
Ceva's theorem In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are kn ...
, which gives a criterion for determining when three such lines are
concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...
. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
: here
Menelaus' theorem Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ''ABC'', and a transversal line that crosses ''BC'', ''AC'', and ''AB'' at points ''D'', ''E'', and ''F'' respec ...
gives a useful general criterion. In this section just a few of the most commonly encountered constructions are explained. A
perpendicular bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
of a side of a triangle is a straight line passing through the
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimens ...
of the side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's
circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
, usually denoted by O; this point is the center of the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
, the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
passing through all three vertices. The diameter of this circle, called the ''circumdiameter'', can be found from the law of sines stated above. The circumcircle's radius is called the ''circumradius''.
Thales' theorem In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
implies that if the circumcenter is located on a side of the triangle, then the opposite angle is a right one. If the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. An
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the ''base'' of the altitude, and the point where the altitude intersects the base (or its extension) is called the ''foot'' of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
of the triangle, usually denoted by H. The orthocenter lies inside the triangle if and only if the triangle is acute. An
angle bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
, usually denoted by I, the center of the triangle's
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
. The incircle is the circle which lies inside the triangle and touches all three sides. Its radius is called the ''inradius''. There are three other important circles, the
excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
s; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an
orthocentric system In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and t ...
. A
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
of a triangle is a straight line through a
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
and the
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimens ...
of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
or geometric barycenter, usually denoted by G. The centroid of a rigid triangular object (cut out of a thin sheet of uniform density) is also its
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
: the object can be balanced on its centroid in a uniform gravitational field. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side. The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's
nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of ea ...
. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the
Feuerbach point In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on th ...
) and the three
excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
s. The orthocenter (blue point), center of the nine-point circle (red), centroid (orange), and circumcenter (green) all lie on a single line, known as
Euler's line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, inclu ...
(red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. The center of the incircle is not in general located on Euler's line. If one reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the
symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corr ...
of the triangle.


Computing the sides and angles

There are various standard methods for calculating the length of a side or the measure of an angle. Certain methods are suited to calculating values in a right-angled triangle; more complex methods may be required in other situations.


Trigonometric ratios in right triangles

In
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
s, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known as follows: * The ''
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
'' is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case ''h''. * The ''opposite side'' is the side opposite to the angle we are interested in, in this case ''a''. * The ''adjacent side'' is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is ''b''.


Sine, cosine and tangent

The ''sine'' of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case :\sin A = \frac = \frac \,. This ratio does not depend on the particular right triangle chosen, as long as it contains the angle ''A'', since all those triangles are similar. The ''cosine'' of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case :\cos A = \frac = \frac \,. The ''tangent'' of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case :\tan A = \frac = \frac =\frac \,. The acronym " SOH-CAH-TOA" is a useful
mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and imag ...
for these ratios.


Inverse functions

The
inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
can be used to calculate the internal angles for a right angled triangle with the length of any two sides. Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse. :\theta = \arcsin \left( \frac \right) Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypotenuse. :\theta = \arccos \left( \frac \right) Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side. :\theta = \arctan \left( \frac \right) In introductory geometry and trigonometry courses, the notation sin−1, cos−1, etc., are often used in place of arcsin, arccos, etc. However, the arcsin, arccos, etc., notation is standard in higher mathematics where trigonometric functions are commonly raised to powers, as this avoids confusion between
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
and
compositional inverse In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\t ...
.


Sine, cosine and tangent rules

The
law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and a ...
, or sine rule, states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is :\frac = \frac = \frac = 2R , where R is the radius of the circumscribed circle of the given triangle. Another interpretation of this theorem is that every triangle with angles α, β and γ is similar to a triangle with side lengths equal to sin α, sin β and sin γ. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. The length of the sides of that triangle will be sin α, sin β and sin γ. The side whose length is sin α is opposite to the angle whose measure is α, etc. The
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
, or cosine rule, connects the length of an unknown side of a triangle to the length of the other sides and the angle opposite to the unknown side. As per the law: For a triangle with length of sides ''a'', ''b'', ''c'' and angles of α, β, γ respectively, given two known lengths of a triangle ''a'' and ''b'', and the angle between the two known sides γ (or the angle opposite to the unknown side ''c''), to calculate the third side ''c'', the following formula can be used: :c^2\ = a^2 + b^2 - 2ab\cos(\gamma) :b^2\ = a^2 + c^2 - 2ac\cos(\beta) :a^2\ = b^2 + c^2 - 2bc\cos(\alpha) If the lengths of all three sides of any triangle are known the three angles can be calculated: :\alpha=\arccos\left(\frac\right) :\beta=\arccos\left(\frac\right) :\gamma=\arccos\left(\frac\right) The
law of tangents In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, , , and are the lengths of the three sides of the triangle, and , , ...
, or tangent rule, can be used to find a side or an angle when two sides and an angle or two angles and a side are known. It states that: :\frac = \frac.


Solution of triangles

"Solution of triangles" is the main
trigonometric Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
problem: to find missing characteristics of a triangle (three angles, the lengths of the three sides etc.) when at least three of these characteristics are given. The triangle can be located on a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
or on a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. This problem often occurs in various trigonometric applications, such as
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
,
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
,
construction Construction is a general term meaning the art and science to form objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and com ...
,
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
etc.


Area

Calculating the area ''T'' of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is: :T=\tfrac12 bh, where ''b'' is the length of the base of the triangle, and ''h'' is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the base onto the line containing the base. In 499 CE
Aryabhata Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
, used this illustrated method in the ''
Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...
'' (section 2.6). Although simple, this formula is only useful if the height can be readily found, which is not always the case. For example, the surveyor of a triangular field might find it relatively easy to measure the length of each side, but relatively difficult to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle.


Using trigonometry

The height of a triangle can be found through the application of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
. ''Knowing SAS'': Using the labels in the image on the right, the altitude is . Substituting this in the formula T=\tfrac12 bh derived above, the area of the triangle can be expressed as: :T = \tfrac12 ab\sin \gamma = \tfrac12 bc\sin \alpha = \tfrac12 ca\sin \beta (where α is the interior angle at ''A'', β is the interior angle at ''B'', \gamma is the interior angle at ''C'' and ''c'' is the line AB). Furthermore, since sin α = sin (''π'' − α) = sin (β + \gamma), and similarly for the other two angles: :T = \tfrac12 ab\sin (\alpha+\beta) = \tfrac12 bc\sin (\beta+\gamma) = \tfrac12 ca\sin (\gamma+\alpha). ''Knowing AAS'': :T = \frac , and analogously if the known side is ''a'' or ''c''. ''Knowing ASA'': :T = \frac = \frac, and analogously if the known side is ''b'' or ''c''.


Heron's formula

The shape of the triangle is determined by the lengths of the sides. Therefore, the area can also be derived from the lengths of the sides. By
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
: :T = \sqrt where s= \tfrac12(a+b+c) is the
semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
, or half of the triangle's perimeter. Three other equivalent ways of writing Heron's formula are :T = \tfrac14 \sqrt :T = \tfrac14 \sqrt :T = \tfrac14 \sqrt.


Using vectors

The area of a
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
embedded in a three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
can be calculated using vectors. Let vectors AB and AC point respectively from ''A'' to ''B'' and from ''A'' to ''C''. The area of parallelogram ''ABDC'' is then :, \mathbf\times\mathbf, , which is the magnitude of the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of vectors AB and AC. The area of triangle ABC is half of this, :\tfrac12, \mathbf\times\mathbf, . The area of triangle ''ABC'' can also be expressed in terms of
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
s as follows: :\tfrac12 \sqrt =\tfrac12 \sqrt.\, In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to (''x''1,''y''1) and AC as (''x''2,''y''2), this can be rewritten as: :\tfrac12, x_1 y_2 - x_2 y_1, .


Using coordinates

If vertex ''A'' is located at the origin (0, 0) of a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
and the coordinates of the other two vertices are given by and , then the area can be computed as times the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
:T = \tfrac12\left, \det\beginx_B & x_C \\ y_B & y_C \end\ = \tfrac12 , x_B y_C - x_C y_B, . For three general vertices, the equation is: :T = \tfrac12 \left, \det\beginx_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1\end \ = \tfrac12 \big, x_A y_B - x_A y_C + x_B y_C - x_B y_A + x_C y_A - x_C y_B \big, , which can be written as :T = \tfrac12 \big, (x_A - x_C) (y_B - y_A) - (x_A - x_B) (y_C - y_A) \big, . If the points are labeled sequentially in the counterclockwise direction, the above determinant expressions are positive and the absolute value signs can be omitted. The above formula is known as the
shoelace formula The shoelace formula, shoelace algorithm, or shoelace method (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian c ...
or the surveyor's formula. If we locate the vertices in the complex plane and denote them in counterclockwise sequence as , , and , and denote their complex conjugates as \bar a, \bar b, and \bar c, then the formula :T=\frac\begina & \bar a & 1 \\ b & \bar b & 1 \\ c & \bar c & 1 \end is equivalent to the shoelace formula. In three dimensions, the area of a general triangle , and ) is the
Pythagorean sum Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...
of the areas of the respective projections on the three principal planes (i.e. ''x'' = 0, ''y'' = 0 and ''z'' = 0): :T = \tfrac12 \sqrt.


Using line integrals

The area within any closed curve, such as a triangle, is given by the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
around the curve of the algebraic or signed distance of a point on the curve from an arbitrary oriented straight line ''L''. Points to the right of ''L'' as oriented are taken to be at negative distance from ''L'', while the weight for the integral is taken to be the component of arc length parallel to ''L'' rather than arc length itself. This method is well suited to computation of the area of an arbitrary
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
. Taking ''L'' to be the ''x''-axis, the line integral between consecutive vertices (''xi'',''yi'') and (''x''''i''+1,''y''''i''+1) is given by the base times the mean height, namely . The sign of the area is an overall indicator of the direction of traversal, with negative area indicating counterclockwise traversal. The area of a triangle then falls out as the case of a polygon with three sides. While the line integral method has in common with other coordinate-based methods the arbitrary choice of a coordinate system, unlike the others it makes no arbitrary choice of vertex of the triangle as origin or of side as base. Furthermore, the choice of coordinate system defined by ''L'' commits to only two degrees of freedom rather than the usual three, since the weight is a local distance (e.g. in the above) whence the method does not require choosing an axis normal to ''L''. When working in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
it is not necessary to convert to
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
to use line integration, since the line integral between consecutive vertices (''ri'',θ''i'') and (''r''''i''+1''i''+1) of a polygon is given directly by . This is valid for all values of θ, with some decrease in numerical accuracy when , θ, is many orders of magnitude greater than π. With this formulation negative area indicates clockwise traversal, which should be kept in mind when mixing polar and cartesian coordinates. Just as the choice of ''y''-axis () is immaterial for line integration in cartesian coordinates, so is the choice of zero heading () immaterial here.


Formulas resembling Heron's formula

Three formulas have the same structure as Heron's formula but are expressed in terms of different variables. First, denoting the medians from sides ''a'', ''b'', and ''c'' respectively as ''ma'', ''mb'', and ''mc'' and their semi-sum as σ, we have :T = \tfrac43 \sqrt. Next, denoting the altitudes from sides ''a'', ''b'', and ''c'' respectively as ''ha'', ''hb'', and ''hc'', and denoting the semi-sum of the reciprocals of the altitudes as H = (h_a^ + h_b^ + h_c^)/2 we have :T^ = 4 \sqrt. And denoting the semi-sum of the angles' sines as , we have :T = D^ \sqrt where ''D'' is the diameter of the circumcircle: D=\tfrac = \tfrac = \tfrac.


Using Pick's theorem

See
Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 18 ...
for a technique for finding the area of any arbitrary lattice polygon (one drawn on a grid with vertically and horizontally adjacent lattice points at equal distances, and with vertices on lattice points). The theorem states: :T = I + \tfrac12 B - 1 where ''I'' is the number of internal lattice points and ''B'' is the number of lattice points lying on the border of the polygon.


Other area formulas

Numerous other area formulas exist, such as :T = r \cdot s, where ''r'' is the
inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
, and ''s'' is the
semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
(in fact, this formula holds for ''all''
tangential polygon In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual poly ...
s), and :T=r_a(s-a)=r_b(s-b)=r_c(s-c) where r_a, \, r_b,\, r_c are the radii of the
excircles In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
tangent to sides ''a, b, c'' respectively. We also have :T = \tfrac12 D^(\sin \alpha)(\sin \beta)(\sin \gamma) and :T = \frac = \frac for circumdiameter ''D''; and :T = \tfrac14(\tan \alpha)(b^+c^-a^) for angle α ≠ 90°. The area can also be expressed as :T = \sqrt. In 1885, Baker gave a collection of over a hundred distinct area formulas for the triangle. These include: :T = \tfrac12\sqrt :T = \tfrac12 \sqrt, :T = \frac, :T = \frac for circumradius (radius of the circumcircle) ''R'', and :T = \frac.


Upper bound on the area

The area ''T'' of any triangle with perimeter ''p'' satisfies :T\le \tfrac, with equality holding if and only if the triangle is equilateral. Other upper bounds on the area ''T'' are given by :4\sqrtT \leq a^2+b^2+c^2 and :4\sqrtT \leq \frac, both again holding if and only if the triangle is equilateral.


Bisecting the area

There are infinitely many lines that bisect the area of a triangle.Dunn, J.A., and Pretty, J.E., "Halving a triangle," ''
Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...
'' 56, May 1972, 105–108.
Three of them are the medians, which are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter. There can be one, two, or three of these for any given triangle.


Further formulas for general Euclidean triangles

The formulas in this section are true for all Euclidean triangles.


Medians, angle bisectors, perpendicular side bisectors, and altitudes

The medians and the sides are related by :\frac(a^+b^+c^)=m_a^+m_b^+m_c^ and :m_a=\frac \sqrt= \sqrt, and equivalently for ''mb'' and ''mc''. For angle A opposite side ''a'', the length of the internal angle bisector is given by :w_A = \frac = \sqrt = \frac\cos \frac, for semiperimeter ''s'', where the bisector length is measured from the vertex to where it meets the opposite side. The interior perpendicular bisectors are given by :p_a=\frac, :p_b=\frac, :p_c=\frac, where the sides are a \ge b \ge c and the area is T. The altitude from, for example, the side of length ''a'' is :h_a = \frac.


Circumradius and inradius

The following formulas involve the circumradius ''R'' and the inradius ''r'': :R = \sqrt; :r = \sqrt; :\frac = \frac + \frac + \frac where ''ha'' etc. are the altitudes to the subscripted sides; :\frac = \frac = \cos \alpha + \cos \beta + \cos \gamma -1;Longuet-Higgins, Michael S., "On the ratio of the inradius to the circumradius of a triangle", ''
Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...
'' 87, March 2003, 119–120.
and :2Rr = \frac. The product of two sides of a triangle equals the altitude to the third side times the diameter ''D'' of the circumcircle:Altshiller-Court, Nathan, ''College Geometry'', Dover, 2007. :ab=h_cD, \quad \quad bc=h_aD, \quad ca=h_bD.


Adjacent triangles

Suppose two adjacent but non-overlapping triangles share the same side of length ''f'' and share the same circumcircle, so that the side of length ''f'' is a chord of the circumcircle and the triangles have side lengths (''a'', ''b'', ''f'') and (''c'', ''d'', ''f''), with the two triangles together forming a
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
with side lengths in sequence (''a'', ''b'', ''c'', ''d''). ThenJohnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ. Co., 2007 :f^2 = \frac. \,


Centroid

Let ''G'' be the centroid of a triangle with vertices ''A'', ''B'', and ''C'', and let ''P'' be any interior point. Then the distances between the points are related by :(PA)^2 + (PB)^2 +(PC)^2 =(GA)^2 + (GB)^2 + (GC)^2 +3(PG)^2. \, The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices: :AB^2+BC^2+CA^2=3(GA^2+GB^2+GC^2). Let ''qa'', ''qb'', and ''qc'' be the distances from the centroid to the sides of lengths ''a'', ''b'', and ''c''. Then : \frac = \frac, \quad \quad \frac = \frac, \quad \quad \frac = \frac \, and :q_a \cdot a = q_b \cdot b = q_c \cdot c = \frac T \, for area ''T''.


Circumcenter, incenter, and orthocenter

Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius. Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle. This method is especially useful for deducing the properties of more abstract forms of triangles, such as the ones induced by
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s, that otherwise have the same properties as usual triangles.
Euler's theorem In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congru ...
states that the distance ''d'' between the circumcenter and the incenter is given by :\displaystyle d^2=R(R-2r) or equivalently :\frac + \frac = \frac, where ''R'' is the circumradius and ''r'' is the inradius. Thus for all triangles ''R'' ≥ 2''r'', with equality holding for equilateral triangles. If we denote that the orthocenter divides one altitude into segments of lengths ''u'' and ''v'', another altitude into segment lengths ''w'' and ''x'', and the third altitude into segment lengths ''y'' and ''z'', then ''uv'' = ''wx'' = ''yz''. The distance from a side to the circumcenter equals half the distance from the opposite vertex to the orthocenter. The sum of the squares of the distances from the vertices to the orthocenter ''H'' plus the sum of the squares of the sides equals twelve times the square of the circumradius: :AH^2+BH^2+CH^2+a^2+b^2+c^2=12R^2.


Angles

In addition to the
law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and a ...
, the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
, the
law of tangents In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, , , and are the lengths of the three sides of the triangle, and , , ...
, and the trigonometric existence conditions given earlier, for any triangle :a=b\cos C+c\cos B, \quad b=c\cos A+a\cos C, \quad c=a\cos B+b\cos A.


Morley's trisector theorem

Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent
angle trisector Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
s form an equilateral triangle, called the Morley triangle.


Figures inscribed in a triangle


Conics

As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Every triangle has a unique
Steiner inellipse In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse i ...
which is interior to the triangle and tangent at the midpoints of the sides.
Marden's theorem In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its deriva ...
shows how to find the foci of this ellipse. This ellipse has the greatest area of any ellipse tangent to all three sides of the triangle. The
Mandart inellipse In geometry, the Mandart inellipse of a triangle is an ellipse inscribed within the triangle, tangent to its sides at the contact points of its excircles (which are also the vertices of the extouch triangle and the endpoints of the splitters). ...
of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles. For any ellipse inscribed in a triangle ''ABC'', let the foci be ''P'' and ''Q''. Then :\frac + \frac + \frac = 1.


Convex polygon

Every
convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
with area ''T'' can be inscribed in a triangle of area at most equal to 2''T''. Equality holds (exclusively) for a
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
.


Hexagon

The
Lemoine hexagon In geometry, the Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. There are two definitions of th ...
is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its
symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corr ...
. In either its simple form or its self-intersecting form, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.


Squares

Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two ''distinct'' inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has side of length ''q''''a'' and the triangle has a side of length ''a'', part of which side coincides with a side of the square, then ''q''''a'', ''a'', the altitude ''h''''a'' from the side ''a'', and the triangle's area ''T'' are related according to :q_a=\frac=\frac. The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when , , and the altitude of the triangle from the base of length ''a'' is equal to ''a''. The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is 2\sqrt/3 = 0.94.... Both of these extreme cases occur for the isosceles right triangle.


Triangles

From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the
pedal triangle In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the thr ...
of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle. The
Gergonne triangle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
or intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle. The
extouch triangle In Euclidean geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle. Coordinates The vertices of the extouch triangle are given in trilinear coordinates by: :\begin T_A &= 0 ...
of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended).


Figures circumscribed about a triangle

The
tangential triangle In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertices. Thus the incircle of the ...
of a reference triangle (other than a right triangle) is the triangle whose sides are on the
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s to the reference triangle's circumcircle at its vertices. As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Further, every triangle has a unique
Steiner circumellipse In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse ( ellipse that touches the triangle at its vertices) whose center is the triangle's ...
, which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area. The
Kiepert hyperbola *Friedrich Wilhelm August Ludwig Kiepert Friedrich Wilhelm August Ludwig Kiepert (6 October 1846 – 5 September 1934) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typic ...
is the unique
conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...
which passes through the triangle's three vertices, its centroid, and its circumcenter. Of all triangles contained in a given
convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
, there exists a triangle with maximal area whose vertices are all vertices of the given polygon.


Specifying the location of a point in a triangle

One way to identify locations of points in (or outside) a triangle is to place the triangle in an arbitrary location and orientation in the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, and to use Cartesian coordinates. While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane. Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which give a congruent triangle, or even by rescaling it to give a similar triangle: *
Trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
specify the relative distances of a point from the sides, so that coordinates x : y : z indicate that the ratio of the distance of the point from the first side to its distance from the second side is x : y , etc. *
Barycentric coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
of the form \alpha :\beta :\gamma specify the point's location by the relative weights that would have to be put on the three vertices in order to balance the otherwise weightless triangle on the given point.


Non-planar triangles

A non-planar triangle is a triangle which is not contained in a (flat) plane. Some examples of non-planar triangles in non-Euclidean geometries are
spherical triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
s in
spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
and
hyperbolic triangle In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three points called ''angles'' or ''vertices''. Just as in the Euclidean case, three po ...
s in
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
. While the measures of the internal angles in planar triangles always sum to 180°, a hyperbolic triangle has measures of angles that sum to less than 180°, and a spherical triangle has measures of angles that sum to more than 180°. A hyperbolic triangle can be obtained by drawing on a negatively curved surface, such as a
saddle surface In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functi ...
, and a spherical triangle can be obtained by drawing on a positively curved surface such as a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. Thus, if one draws a giant triangle on the surface of the Earth, one will find that the sum of the measures of its angles is greater than 180°; in fact it will be between 180° and 540°.Watkins, Matthew, ''Useful Mathematical and Physical Formulae'', Walker and Co., 2000. In particular it is possible to draw a triangle on a sphere such that the measure of each of its internal angles is equal to 90°, adding up to a total of 270°. Specifically, on a sphere the sum of the angles of a triangle is :180° × (1 + 4''f''), where ''f'' is the fraction of the sphere's area which is enclosed by the triangle. For example, suppose that we draw a triangle on the Earth's surface with vertices at the North Pole, at a point on the equator at
0° longitude The IERS Reference Meridian (IRM), also called the International Reference Meridian, is the prime meridian (0° longitude) maintained by the International Earth Rotation and Reference Systems Service (IERS). It passes about 5.3 arcseconds east o ...
, and a point on the equator at 90° West longitude. The
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
line between the latter two points is the equator, and the great circle line between either of those points and the
North Pole The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Mag ...
is a line of longitude; so there are right angles at the two points on the equator. Moreover, the angle at the North Pole is also 90° because the other two vertices differ by 90° of longitude. So the sum of the angles in this triangle is . The triangle encloses 1/4 of the northern hemisphere (90°/360° as viewed from the North Pole) and therefore 1/8 of the Earth's surface, so in the formula ; thus the formula correctly gives the sum of the triangle's angles as 270°. From the above angle sum formula we can also see that the Earth's surface is locally flat: If we draw an arbitrarily small triangle in the neighborhood of one point on the Earth's surface, the fraction ''f'' of the Earth's surface which is enclosed by the triangle will be arbitrarily close to zero. In this case the angle sum formula simplifies to 180°, which we know is what Euclidean geometry tells us for triangles on a flat surface.


Triangles in construction

Rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
s have been the most popular and common geometric form for buildings since the shape is easy to stack and organize; as a standard, it is easy to design furniture and fixtures to fit inside rectangularly shaped buildings. But triangles, while more difficult to use conceptually, provide a great deal of strength. As computer technology helps
architect An architect is a person who plans, designs and oversees the construction of buildings. To practice architecture means to provide services in connection with the design of buildings and the space within the site surrounding the buildings that h ...
s design creative new buildings, triangular shapes are becoming increasingly prevalent as parts of buildings and as the primary shape for some types of skyscrapers as well as building materials. In Tokyo in 1989, architects had wondered whether it was possible to build a 500-story tower to provide affordable office space for this densely packed city, but with the danger to buildings from
earthquake An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
s, architects considered that a triangular shape would be necessary if such a building were to be built. In
New York City New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the L ...
, as
Broadway Broadway may refer to: Theatre * Broadway Theatre (disambiguation) * Broadway theatre, theatrical productions in professional theatres near Broadway, Manhattan, New York City, U.S. ** Broadway (Manhattan), the street **Broadway Theatre (53rd Stree ...
crisscrosses major avenues, the resulting blocks are cut like triangles, and buildings have been built on these shapes; one such building is the triangularly shaped
Flatiron Building The Flatiron Building, originally the Fuller Building, is a triangular 22-story, steel-framed landmarked building at 175 Fifth Avenue in the eponymous Flatiron District neighborhood of the Boroughs of New York City, borough of Manhattan in New ...
which real estate people admit has a "warren of awkward spaces that do not easily accommodate modern office furniture" but that has not prevented the structure from becoming a landmark icon. Designers have made houses in
Norway Norway, officially the Kingdom of Norway, is a Nordic country in Northern Europe, the mainland territory of which comprises the western and northernmost portion of the Scandinavian Peninsula. The remote Arctic island of Jan Mayen and t ...
using triangular themes. Triangle shapes have appeared in churches as well as public buildings including colleges as well as supports for innovative home designs. Triangles are sturdy; while a rectangle can collapse into a
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
from pressure to one of its points, triangles have a natural strength which supports structures against lateral pressures. A triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports the other two. A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense. Some innovative designers have proposed making
bricks A brick is a type of block used to build walls, pavements and other elements in masonry construction. Properly, the term ''brick'' denotes a block composed of dried clay, but is now also used informally to denote other chemically cured cons ...
not out of rectangles, but with triangular shapes which can be combined in three dimensions. It is likely that triangles will be used increasingly in new ways as architecture increases in complexity. It is important to remember that triangles are strong in terms of rigidity, but while packed in a
tessellating A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
arrangement triangles are not as strong as
hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...
s under compression (hence the prevalence of hexagonal forms in
nature Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ...
). Tessellated triangles still maintain superior strength for
cantilever A cantilever is a rigid structural element that extends horizontally and is supported at only one end. Typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. Like other structural elements, a canti ...
ing however, and this is the basis for one of the strongest man made structures, the tetrahedral truss.


See also

*
Apollonius' theorem In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, t ...
*
Congruence (geometry) In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be ...
*
Desargues' theorem In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and tho ...
*
Dragon's Eye (symbol) Dragon's Eye may refer to: * Dragon's Eye (symbol), an ancient geometric tetrahedron or triangle * ''Dragon's Eye'' (TV programme), a BBC Cymru Wales television programme * '' Dragonseye'' (or ''Red Star Rising''), a science fiction novel by Anne ...
*
Fermat point In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest ...
*
Hadwiger–Finsler inequality In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :a^ + b^ + c^ \geq (a - b)^ ...
*
Heronian triangle In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula. Heron's formula implies ...
*
Integer triangle An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational Rationality is the quality of being guided by or based on re ...
*
Law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
*
Law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and a ...
*
Law of tangents In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, , , and are the lengths of the three sides of the triangle, and , , ...
*
Lester's theorem In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle. The result is named after June Lester, who published it in 1997, and t ...
*
List of triangle inequalities In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of th ...
*
List of triangle topics This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal's triangle or triangular matrix, triangular matrices, or concretely in ...
*
Modern triangle geometry In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and ...
*
Ono's inequality In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by T. Ono in 1914, the inequality is actually false; however, the statement is true for acute triangles and right triangles, ...
*
Pedal triangle In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the thr ...
*
Pedoe's inequality In geometry, Pedoe's inequality (also Neuberg–Pedoe inequality), named after Daniel Pedoe (1910–1998) and Joseph Jean Baptiste Neuberg (1840–1926), states that if ''a'', ''b'', and ''c'' are the lengths of the sides of a triangle with area ''& ...
*
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
*
Special right triangles A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...
*
Triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...
*
Triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
*
Triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy categ ...
*
Triangulation (topology) In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangula ...


Notes


References


External links

* * Clark Kimberling
Encyclopedia of triangle centers
Lists some 5200 interesting points associated with any triangle. {{Authority control