A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C displaystyle triangle ABC .
In
Contents 1 Types of triangle 1.1 By lengths of sides 1.2 By internal angles 2 Basic facts 2.1 Similarity and congruence 2.2 Right triangles 3 Existence of a triangle 3.1 Condition on the sides 3.2 Conditions on the angles 3.2.1 Trigonometric conditions 4 Points, lines, and circles associated with a triangle 5 Computing the sides and angles 5.1 Trigonometric ratios in right triangles 5.1.1 Sine, cosine and tangent 5.1.2 Inverse functions 5.2 Sine, cosine and tangent rules 5.3 Solution of triangles 6 Computing the area of a triangle 6.1 Using trigonometry 6.2 Using Heron's formula 6.3 Using vectors 6.4 Using coordinates 6.5 Using line integrals 6.6 Formulas resembling Heron's formula 6.7 Using Pick's theorem 6.8 Other area formulas 6.9 Upper bound on the area 6.10 Bisecting the area 7 Further formulas for general Euclidean triangles 7.1 Medians, angle bisectors, perpendicular side bisectors, and
altitudes
7.2
8 Morley's trisector theorem 9 Figures inscribed in a triangle 9.1 Conics 9.2 Convex polygon 9.3 Hexagon 9.4 Squares 9.5 Triangles 10 Figures circumscribed about a triangle 11 Specifying the location of a point in a triangle 12 Non-planar triangles 13 Triangles in construction 14 See also 15 Notes 16 References 17 External links Types of triangle
By lengths of sides Triangles can be classified according to the lengths of their sides: An equilateral triangle has all sides the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.[1] An isosceles triangle has two sides of equal length.[note 1][2] 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, which was known by Euclid. 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.[2] The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths.[3] Equivalently, it has all angles of different measure. Equilateral Isosceles Scalene Hatch marks, 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; 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 angles, measured here in degrees. A right triangle (or right-angled triangle, formerly called a
rectangled triangle) has one of its interior angles measuring 90° (a
right angle). The side opposite to the right angle is the hypotenuse,
the longest side of the triangle. The other two sides are called the
legs or catheti[4] (singular: cathetus) of the triangle. Right
triangles obey the Pythagorean theorem: the sum of the squares of the
lengths of the two legs is equal to the square of the length of the
hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs
and c is the length of the hypotenuse.
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 such a triangle is therefore equilateral. Right Obtuse Acute
⏟ displaystyle underbrace qquad qquad qquad qquad qquad qquad _ Oblique Basic facts A triangle, showing exterior angle d. Triangles are assumed to be two-dimensional plane figures, unless the
context provides otherwise (see Non-planar triangles, below). In
rigorous treatments, a triangle is therefore called a 2-simplex (see
also Polytope). Elementary facts about triangles were presented by
The measures of the interior angles of the triangle always add up to 180 degrees (same color to point out they are equal). The sum of the measures of the interior angles of a triangle in
Two triangles that are congruent have exactly the same size and shape:[note 4] 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 conditions 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-
An important condition is: Side-Side-
Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry. Right triangles The Pythagorean theorem A central theorem is the Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse 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 . displaystyle 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 + b + 90 ∘ = 180 ∘ ⇒ a + b = 90 ∘ ⇒ a = 90 ∘ − b . displaystyle 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 √2. 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 √3: c = 2 a displaystyle c=2a, b = a × 3 . displaystyle b=atimes sqrt 3 . For all triangles, angles and sides are related by the law of cosines and law of sines (also called the cosine rule and sine rule). Existence of a triangle Condition on the sides The triangle inequality 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 any one of the following conditions holds: tan α 2 tan β 2 + tan β 2 tan γ 2 + tan γ 2 tan α 2 = 1 , displaystyle tan frac alpha 2 tan frac beta 2 +tan frac beta 2 tan frac gamma 2 +tan frac gamma 2 tan frac alpha 2 =1, [6] sin 2 α 2 + sin 2 β 2 + sin 2 γ 2 + 2 sin α 2 sin β 2 sin γ 2 = 1 , displaystyle sin ^ 2 frac alpha 2 +sin ^ 2 frac beta 2 +sin ^ 2 frac gamma 2 +2sin frac alpha 2 sin frac beta 2 sin frac gamma 2 =1, [6] sin ( 2 α ) + sin ( 2 β ) + sin ( 2 γ ) = 4 sin ( α ) sin ( β ) sin ( γ ) , displaystyle sin(2alpha )+sin(2beta )+sin(2gamma )=4sin(alpha )sin(beta )sin(gamma ), cos 2 α + cos 2 β + cos 2 γ + 2 cos ( α ) cos ( β ) cos ( γ ) = 1 , displaystyle cos ^ 2 alpha +cos ^ 2 beta +cos ^ 2 gamma +2cos(alpha )cos(beta )cos(gamma )=1, [7] tan ( α ) + tan ( β ) + tan ( γ ) = tan ( α ) tan ( β ) tan ( γ ) , displaystyle 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
The circumcenter is the center of a circle passing through the three vertices of the triangle. A perpendicular bisector of a side of a triangle is a straight line
passing through the midpoint 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, usually
denoted by O; this point is the center of the circumcircle, the circle
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.
The intersection of the altitudes is the orthocenter. An altitude 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 of the triangle, usually denoted by H. The orthocenter lies inside the triangle if and only if the triangle is acute. The intersection of the angle bisectors is the center of the incircle. An angle bisector 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, usually denoted by I, the center of the triangle's incircle. 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 excircles; 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. The intersection of the medians is the centroid. A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid 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: 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. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles.
The centroid (yellow), orthocenter (blue), circumcenter (green) and
center of the nine-point circle (red point) all lie on a single line,
known as
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 Main article: Trigonometric functions A right triangle always includes a 90° (π/2 radians) angle, here
with label C. Angles A and B may vary.
In right triangles, 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 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 = opposite side hypotenuse = a h . displaystyle sin A= frac text opposite side text hypotenuse = frac a h ,. 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 = adjacent side hypotenuse = b h . displaystyle cos A= frac text adjacent side text hypotenuse = frac b h ,. 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 = opposite side adjacent side = a b = sin A cos A . displaystyle tan A= frac text opposite side text adjacent side = frac a b = frac sin A cos A ,. The acronym "SOH-CAH-TOA" is a useful mnemonic for these ratios. Inverse functions The inverse trigonometric functions 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. θ = arcsin ( opposite side hypotenuse ) displaystyle theta =arcsin left( frac text opposite side text hypotenuse right) Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypotenuse. θ = arccos ( adjacent side hypotenuse ) displaystyle theta =arccos left( frac text adjacent side text hypotenuse right) Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side. θ = arctan ( opposite side adjacent side ) displaystyle theta =arctan left( frac text opposite side text adjacent side 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 and compositional inverse. Sine, cosine and tangent rules Main articles: Law of sines, Law of cosines, and Law of tangents A triangle with sides of length a, b and c and angles of α, β and γ respectively. The law of sines, or sine rule,[8] states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is a sin α = b sin β = c sin γ . displaystyle frac a sin alpha = frac b sin beta = frac c sin gamma . This ratio is equal to the diameter 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, 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.[8] 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 − 2 a b cos ( γ ) displaystyle c^ 2 =a^ 2 +b^ 2 -2abcos(gamma ) b 2 = a 2 + c 2 − 2 a c cos ( β ) displaystyle b^ 2 =a^ 2 +c^ 2 -2accos(beta ) a 2 = b 2 + c 2 − 2 b c cos ( α ) displaystyle a^ 2 =b^ 2 +c^ 2 -2bccos(alpha ) If the lengths of all three sides of any triangle are known the three angles can be calculated: α = arccos ( b 2 + c 2 − a 2 2 b c ) displaystyle alpha =arccos left( frac b^ 2 +c^ 2 -a^ 2 2bc right) β = arccos ( a 2 + c 2 − b 2 2 a c ) displaystyle beta =arccos left( frac a^ 2 +c^ 2 -b^ 2 2ac right) γ = arccos ( a 2 + b 2 − c 2 2 a b ) displaystyle gamma =arccos left( frac a^ 2 +b^ 2 -c^ 2 2ab right) The law of tangents, 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:[9] a − b a + b = tan [ 1 2 ( α − β ) ] tan [ 1 2 ( α + β ) ] . displaystyle frac a-b a+b = frac tan[ frac 1 2 (alpha -beta )] tan[ frac 1 2 (alpha +beta )] . Solution of triangles Main article: Solution of triangles "Solution of triangles" is the main trigonometric 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 or on a sphere. This problem often occurs in various trigonometric applications, such as geodesy, astronomy, construction, navigation etc. Computing the area of a triangle The area of a triangle can be demonstrated as half of the area of a parallelogram which has the same base length and height. 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 = 1 2 b h displaystyle T= frac 1 2 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 side onto the line containing the side itself. In 499 CE
Aryabhata, a great mathematician-astronomer from the classical age of
Applying trigonometry to find the altitude h. The height of a triangle can be found through the application of trigonometry. Knowing SAS: Using the labels in the image on the right, the altitude is h = a sin γ displaystyle gamma . Substituting this in the formula T = 1 2 b h displaystyle T= frac 1 2 bh derived above, the area of the triangle can be expressed as: T = 1 2 a b sin γ = 1 2 b c sin α = 1 2 c a sin β displaystyle T= frac 1 2 absin gamma = frac 1 2 bcsin alpha = frac 1 2 casin beta (where α is the interior angle at A, β is the interior angle at B, γ displaystyle gamma is the interior angle at C and c is the line AB). Furthermore, since sin α = sin (π − α) = sin (β + γ displaystyle gamma ), and similarly for the other two angles: T = 1 2 a b sin ( α + β ) = 1 2 b c sin ( β + γ ) = 1 2 c a sin ( γ + α ) . displaystyle T= frac 1 2 absin(alpha +beta )= frac 1 2 bcsin(beta +gamma )= frac 1 2 casin(gamma +alpha ). Knowing AAS: T = b 2 ( sin α ) ( sin ( α + β ) ) 2 sin β , displaystyle T= frac b^ 2 (sin alpha )(sin(alpha +beta )) 2sin beta , and analogously if the known side is a or c. Knowing ASA:[11] T = a 2 2 ( cot β + cot γ ) = a 2 ( sin β ) ( sin γ ) 2 sin ( β + γ ) , displaystyle T= frac a^ 2 2(cot beta +cot gamma ) = frac a^ 2 (sin beta )(sin gamma ) 2sin(beta +gamma ) , and analogously if the known side is b or c. Using 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: T = s ( s − a ) ( s − b ) ( s − c ) displaystyle T= sqrt s(s-a)(s-b)(s-c) where s = a + b + c 2 displaystyle s= tfrac a+b+c 2 is the semiperimeter, or half of the triangle's perimeter.
Three other equivalent ways of writing
T = 1 4 ( a 2 + b 2 + c 2 ) 2 − 2 ( a 4 + b 4 + c 4 ) displaystyle T= frac 1 4 sqrt (a^ 2 +b^ 2 +c^ 2 )^ 2 -2(a^ 4 +b^ 4 +c^ 4 ) T = 1 4 2 ( a 2 b 2 + a 2 c 2 + b 2 c 2 ) − ( a 4 + b 4 + c 4 ) displaystyle T= frac 1 4 sqrt 2(a^ 2 b^ 2 +a^ 2 c^ 2 +b^ 2 c^ 2 )-(a^ 4 +b^ 4 +c^ 4 ) T = 1 4 ( a + b − c ) ( a − b + c ) ( − a + b + c ) ( a + b + c ) . displaystyle T= frac 1 4 sqrt (a+b-c)(a-b+c)(-a+b+c)(a+b+c) . Using vectors The area of a parallelogram embedded in a three-dimensional Euclidean space 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
A B × A C
, displaystyle mathbf AB times mathbf AC , which is the magnitude of the cross product of vectors AB and AC. The area of triangle ABC is half of this, 1 2
A B × A C
. displaystyle frac 1 2 mathbf AB times mathbf AC . The area of triangle ABC can also be expressed in terms of dot products as follows: 1 2 ( A B ⋅ A B ) ( A C ⋅ A C ) − ( A B ⋅ A C ) 2 = 1 2
A B
2
A C
2 − ( A B ⋅ A C ) 2 . displaystyle frac 1 2 sqrt (mathbf AB cdot mathbf AB )(mathbf AC cdot mathbf AC )-(mathbf AB cdot mathbf AC )^ 2 = frac 1 2 sqrt mathbf AB ^ 2 mathbf AC ^ 2 -(mathbf AB cdot mathbf AC )^ 2 ., In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to (x1,y1) and AC as (x2,y2), this can be rewritten as: 1 2
x 1 y 2 − x 2 y 1
. displaystyle frac 1 2 ,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 and the coordinates of the other two vertices are given by B = (xB, yB) and C = (xC, yC), then the area can be computed as 1⁄2 times the absolute value of the determinant T = 1 2
det ( x B x C y B y C )
= 1 2
x B y C − x C y B
. displaystyle T= frac 1 2 leftdet begin pmatrix x_ B &x_ C \y_ B &y_ C end pmatrix right= frac 1 2 x_ B y_ C -x_ C y_ B . For three general vertices, the equation is: T = 1 2
det ( x A x B x C y A y B y C 1 1 1 )
= 1 2
x A y B − x A y C + x B y C − x B y A + x C y A − x C y B
, displaystyle T= frac 1 2 leftdet begin pmatrix x_ A &x_ B &x_ C \y_ A &y_ B &y_ C \1&1&1end pmatrix right= frac 1 2 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 = 1 2
( x A − x C ) ( y B − y A ) − ( x A − x B ) ( y C − y A )
. displaystyle T= frac 1 2 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.[12] The above formula is known as the shoelace formula or the surveyor's formula. If we locate the vertices in the complex plane and denote them in counterclockwise sequence as a = xA + yAi, b = xB + yBi, and c = xC + yCi, and denote their complex conjugates as a ¯ displaystyle bar a , b ¯ displaystyle bar b , and c ¯ displaystyle bar c , then the formula T = i 4
a a ¯ 1 b b ¯ 1 c c ¯ 1
displaystyle T= frac i 4 begin vmatrix a& bar a &1\b& bar b &1\c& bar c &1end vmatrix is equivalent to the shoelace formula.
In three dimensions, the area of a general triangle A = (xA, yA, zA),
B = (xB, yB, zB) and C = (xC, yC, zC) is the
T = 1 2
x A x B x C y A y B y C 1 1 1
2 +
y A y B y C z A z B z C 1 1 1
2 +
z A z B z C x A x B x C 1 1 1
2 . displaystyle T= frac 1 2 sqrt begin vmatrix x_ A &x_ B &x_ C \y_ A &y_ B &y_ C \1&1&1end vmatrix ^ 2 + begin vmatrix y_ A &y_ B &y_ C \z_ A &z_ B &z_ C \1&1&1end vmatrix ^ 2 + begin vmatrix z_ A &z_ B &z_ C \x_ A &x_ B &x_ C \1&1&1end vmatrix ^ 2 . Using line integrals
The area within any closed curve, such as a triangle, is given by the
line integral 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. Taking L to be the x-axis, the line integral between
consecutive vertices (xi,yi) and (xi+1,yi+1) is given by the base
times the mean height, namely (xi+1 − xi)(yi + yi+1)/2. 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. xi+1 − xi in the above) whence the method does not require
choosing an axis normal to L.
When working in polar coordinates it is not necessary to convert to
T = 4 3 σ ( σ − m a ) ( σ − m b ) ( σ − m c ) . displaystyle T= frac 4 3 sqrt sigma (sigma -m_ a )(sigma -m_ b )(sigma -m_ c ) . 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 − 1 + h b − 1 + h c − 1 ) / 2 displaystyle H=(h_ a ^ -1 +h_ b ^ -1 +h_ c ^ -1 )/2 we have[14] T − 1 = 4 H ( H − h a − 1 ) ( H − h b − 1 ) ( H − h c − 1 ) . displaystyle T^ -1 =4 sqrt H(H-h_ a ^ -1 )(H-h_ b ^ -1 )(H-h_ c ^ -1 ) . And denoting the semi-sum of the angles' sines as S = [(sin α) + (sin β) + (sin γ)]/2, we have[15] T = D 2 S ( S − sin α ) ( S − sin β ) ( S − sin γ ) displaystyle T=D^ 2 sqrt S(S-sin alpha )(S-sin beta )(S-sin gamma ) where D is the diameter of the circumcircle: D = a sin α = b sin β = c sin γ . displaystyle D= tfrac a sin alpha = tfrac b sin beta = tfrac c sin gamma . Using Pick's theorem
See
T = I + 1 2 B − 1 displaystyle T=I+ frac 1 2 B-1 where I displaystyle 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 ⋅ s , displaystyle T=rcdot s, where r is the inradius, and s is the semiperimeter (in fact this formula holds for all tangential polygons), and[16]:Lemma 2 T = r a ( s − a ) = r b ( s − b ) = r c ( s − c ) displaystyle T=r_ a (s-a)=r_ b (s-b)=r_ c (s-c) where r a , r b , r c displaystyle r_ a ,,r_ b ,,r_ c are the radii of the excircles tangent to sides a, b, c respectively. We also have T = 1 2 D 2 ( sin α ) ( sin β ) ( sin γ ) displaystyle T= frac 1 2 D^ 2 (sin alpha )(sin beta )(sin gamma ) and[17] T = a b c 2 D = a b c 4 R displaystyle T= frac abc 2D = frac abc 4R for circumdiameter D; and[18] T = tan α 4 ( b 2 + c 2 − a 2 ) displaystyle T= frac tan alpha 4 (b^ 2 +c^ 2 -a^ 2 ) for angle α ≠ 90°. The area can also be expressed as[19] T = r r a r b r c . displaystyle T= sqrt rr_ a r_ b r_ c . In 1885, Baker[20] gave a collection of over a hundred distinct area formulas for the triangle. These include: T = 1 2 [ a b c h a h b h c ] 1 / 3 , displaystyle T= frac 1 2 [abch_ a h_ b h_ c ]^ 1/3 , T = 1 2 a b h a h b , displaystyle T= frac 1 2 sqrt abh_ a h_ b , T = a + b 2 ( h a − 1 + h b − 1 ) , displaystyle T= frac a+b 2(h_ a ^ -1 +h_ b ^ -1 ) , T = R h b h c a displaystyle T= frac Rh_ b h_ c a for circumradius (radius of the circumcircle) R, and T = h a h b 2 sin γ . displaystyle T= frac h_ a h_ b 2sin gamma . Upper bound on the area The area T of any triangle with perimeter p satisfies T ≤ p 2 12 3 , displaystyle Tleq tfrac p^ 2 12 sqrt 3 , with equality holding if and only if the triangle is equilateral.[21][22]:657 Other upper bounds on the area T are given by[23]:p.290 4 3 T ≤ a 2 + b 2 + c 2 displaystyle 4 sqrt 3 Tleq a^ 2 +b^ 2 +c^ 2 and 4 3 T ≤ 9 a b c a + b + c , displaystyle 4 sqrt 3 Tleq frac 9abc a+b+c , 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.[24] 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
See also: List of triangle inequalities
The formulas in this section are true for all Euclidean triangles.
Medians, angle bisectors, perpendicular side bisectors, and altitudes
Main articles: Median (geometry),
3 4 ( a 2 + b 2 + c 2 ) = m a 2 + m b 2 + m c 2 displaystyle frac 3 4 (a^ 2 +b^ 2 +c^ 2 )=m_ a ^ 2 +m_ b ^ 2 +m_ c ^ 2 and m a = 1 2 2 b 2 + 2 c 2 − a 2 = 1 2 ( a 2 + b 2 + c 2 ) − 3 4 a 2 displaystyle m_ a = frac 1 2 sqrt 2b^ 2 +2c^ 2 -a^ 2 = sqrt frac 1 2 (a^ 2 +b^ 2 +c^ 2 )- frac 3 4 a^ 2 , and equivalently for mb and mc. For angle A opposite side a, the length of the internal angle bisector is given by[26] w A = 2 b c s ( s − a ) b + c = b c [ 1 − a 2 ( b + c ) 2 ] = 2 b c b + c cos A 2 , displaystyle w_ A = frac 2 sqrt bcs(s-a) b+c = sqrt bcleft[1- frac a^ 2 (b+c)^ 2 right] = frac 2bc b+c cos frac A 2 , 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 = 2 a T a 2 + b 2 − c 2 , displaystyle p_ a = frac 2aT a^ 2 +b^ 2 -c^ 2 , p b = 2 b T a 2 + b 2 − c 2 , displaystyle p_ b = frac 2bT a^ 2 +b^ 2 -c^ 2 , p c = 2 c T a 2 − b 2 + c 2 , displaystyle p_ c = frac 2cT a^ 2 -b^ 2 +c^ 2 , where the sides are a ≥ b ≥ c displaystyle ageq bgeq c and the area is T . displaystyle T. [27]:Thm 2 The altitude from, for example, the side of length a is h a = 2 T a . displaystyle h_ a = frac 2T a .
R = a 2 b 2 c 2 ( a + b + c ) ( − a + b + c ) ( a − b + c ) ( a + b − c ) ; displaystyle R= sqrt frac a^ 2 b^ 2 c^ 2 (a+b+c)(-a+b+c)(a-b+c)(a+b-c) ; r = ( − a + b + c ) ( a − b + c ) ( a + b − c ) 4 ( a + b + c ) ; displaystyle r= sqrt frac (-a+b+c)(a-b+c)(a+b-c) 4(a+b+c) ; 1 r = 1 h a + 1 h b + 1 h c displaystyle frac 1 r = frac 1 h_ a + frac 1 h_ b + frac 1 h_ c where ha etc. are the altitudes to the subscripted sides;[25]:p.79 r R = 4 T 2 s a b c = cos α + cos β + cos γ − 1 ; displaystyle frac r R = frac 4T^ 2 sabc =cos alpha +cos beta +cos gamma -1; [7] and 2 R r = a b c a + b + c displaystyle 2Rr= frac abc a+b+c . The product of two sides of a triangle equals the altitude to the third side times the diameter D of the circumcircle:[25]:p.64 a b = h c D , b c = h a D , c a = h b D . displaystyle ab=h_ c D,quad quad bc=h_ a D,quad ca=h_ b D. 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 with side lengths in sequence (a, b, c, d). Then[28]:84 f 2 = ( a c + b d ) ( a d + b c ) ( a b + c d ) . displaystyle f^ 2 = frac (ac+bd)(ad+bc) (ab+cd) ., Centroid Main article: 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[28]:174 ( P A ) 2 + ( P B ) 2 + ( P C ) 2 = ( G A ) 2 + ( G B ) 2 + ( G C ) 2 + 3 ( P G ) 2 . displaystyle (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: A B 2 + B C 2 + C A 2 = 3 ( G A 2 + G B 2 + G C 2 ) . displaystyle AB^ 2 +BC^ 2 +CA^ 2 =3(GA^ 2 +GB^ 2 +GC^ 2 ). [29] Let qa, qb, and qc be the distances from the centroid to the sides of lengths a, b, and c. Then[28]:173 q a q b = b a , q b q c = c b , q a q c = c a displaystyle frac q_ a q_ b = frac b a ,quad quad frac q_ b q_ c = frac c b ,quad quad frac q_ a q_ c = frac c a , and q a ⋅ a = q b ⋅ b = q c ⋅ c = 2 3 T displaystyle q_ a cdot a=q_ b cdot b=q_ c cdot c= frac 2 3 T, for area T.
Circumcenter, incenter, and orthocenter
Main articles: Circumcenter, Incenter, and Orthocenter
Carnot's
d 2 = R ( R − 2 r ) displaystyle displaystyle d^ 2 =R(R-2r) or equivalently 1 R − d + 1 R + d = 1 r , displaystyle frac 1 R-d + frac 1 R+d = frac 1 r , where R is the circumradius and r is the inradius. Thus for all triangles R ≥ 2r, 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.[25]:p.94 The distance from a side to the circumcenter equals half the distance from the opposite vertex to the orthocenter.[25]:p.99 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:[25]:p.102 A H 2 + B H 2 + C H 2 + a 2 + b 2 + c 2 = 12 R 2 . displaystyle AH^ 2 +BH^ 2 +CH^ 2 +a^ 2 +b^ 2 +c^ 2 =12R^ 2 . Angles In addition to the law of sines, the law of cosines, the law of tangents, and the trigonometric existence conditions given earlier, for any triangle a = b cos C + c cos B , b = c cos A + a cos C , c = a cos B + b cos A . displaystyle a=bcos C+ccos B,quad b=ccos A+acos C,quad c=acos B+bcos A. Morley's trisector theorem Main article: Morley's trisector theorem The Morley triangle, resulting from the trisection of each interior angle. This is an example of a finite subdivision rule.
P A ¯ ⋅ Q A ¯ C A ¯ ⋅ A B ¯ + P B ¯ ⋅ Q B ¯ A B ¯ ⋅ B C ¯ + P C ¯ ⋅ Q C ¯ B C ¯ ⋅ C A ¯ = 1. displaystyle frac overline PA cdot overline QA overline CA cdot overline AB + frac overline PB cdot overline QB overline AB cdot overline BC + frac overline PC cdot overline QC overline BC cdot overline CA =1. Convex polygon
Every convex polygon with area T can be inscribed in a triangle of
area at most equal to 2T. Equality holds (exclusively) for a
parallelogram.[32]
Hexagon
The
q a = 2 T a a 2 + 2 T = a h a a + h a . displaystyle q_ a = frac 2Ta a^ 2 +2T = frac ah_ a a+h_ a . The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when a2 = 2T, q = a/2, 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 2 / 3 = 0.94.... displaystyle 2 sqrt 2 /3=0.94.... [34] 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 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
x : y : z displaystyle 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 displaystyle x:y , etc. Barycentric coordinates of the form α : β : γ displaystyle 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 triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry. 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, and a spherical triangle can be obtained by drawing on a positively curved surface such as a sphere. 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°.[36] 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 + 4f), 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, and a point on the equator at 90° West longitude. The
great circle line between the latter two points is the equator, and
the great circle line between either of those points and the North
Pole 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 90° + 90° + 90° = 270°.
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 f = 1/8; 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
The
Rectangles 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 architects 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
earthquakes, architects considered that a triangular shape would have
been necessary if such a building was ever to have been built (it
hasn't by 2011).[37]
In New York City, as Broadway 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 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.[38] Designers have made houses in
Apollonius' theorem Congruence (geometry) Desargues' theorem Dragon's Eye (symbol) Fermat point Hadwiger–Finsler inequality Heronian triangle Integer triangle Law of cosines Law of sines Law of tangents Lester's theorem List of triangle inequalities List of triangle topics Ono's inequality Pedal triangle
Pedoe's inequality
Pythagorean theorem
Notes ^
References ^ Weisstein, Eric W. "Equilateral Triangle". MathWorld.
^ a b Weisstein, Eric W. "Isosceles Triangle". MathWorld.
^ Weisstein, Eric W. "Scalene triangle". MathWorld.
^ Zeidler, Eberhard (2004). Oxford Users' Guide to Mathematics. Oxford
University Press. p. 729. ISBN 978-0-19-850763-5.
^ "Euclid's Elements, Book I, Proposition 32".
^ a b Vardan Verdiyan & Daniel Campos Salas, "Simple trigonometric
substitutions with broad results", Mathematical Reflections no 6,
2007.
^ a b Longuet-Higgins, Michael S., "On the ratio of the inradius to
the circumradius of a triangle",
External links Wikimedia Commons has media related to Triangles. Look up triangle in Wiktionary, the free dictionary. Ivanov, A.B. (2001) [1994], "Triangle", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Clark Kimberling: Encyclopedia of triangle centers. Lists some 5200 interesting points associated with any triangle. v t e Polygons Regular List 1–10 sides Monogon Digon Triangle Equilateral Isosceles Quadrilateral Square Rectangle Rhombus Parallelogram Trapezoid Kite Pentagon
Hexagon
Heptagon
Octagon
11–20 sides Hendecagon Dodecagon Tridecagon Tetradecagon Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon 21–100 sides (selected)
>100 sides 120-gon
257-gon
360-gon
Star polygons (5–12 sides) Pentagram Hexagram Heptagram Octagram Enneagram Decagram Hend |