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A special right triangle is 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 ...
with some regular feature that makes calculations on the
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
easier, or for which simple formulas exist. For example, a right triangle may have
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 ...
s that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in
geometric 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 ca ...
problems without resorting to more advanced methods.


Angle-based

"Angle-based" special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s, is equal to the sum of the other two angles. The side lengths are generally deduced from the basis of the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
or other
geometric 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 ca ...
methods. This approach may be used to rapidly reproduce the values of
trigonometric functions 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 ...
for the angles 30°, 45°, and 60°. Special triangles are used to aid in calculating common trigonometric functions, as below: The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the
equilateral 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 ...
/equiangular (60°–60°–60°) triangle are the three
Möbius triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined mor ...
s in the plane, meaning that they
tessellate A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
the plane via reflections in their sides; see
Triangle group In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle ...
.


45°–45°–90° triangle

In
plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, constructing the diagonal of a
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or radians. Hence, the angles respectively measure 45° (), 45° (), and 90° (). The sides in this triangle are in the ratio 1 : 1 : , which follows immediately from 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 ...
. Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio 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 ...
to the sum of the legs, namely .Posamentier, Alfred S., and Lehman, Ingmar. ''
The Secrets of Triangles ''The Secrets of Triangles: A Mathematical Journey'' is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and , and published in 2012 by Prometheus Books. Topics The book consists of ten chapters, ...
''. Prometheus Books, 2012.
and the greatest ratio of the
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 ...
from the hypotenuse to the sum of the legs, namely . Triangles with these angles are the only possible right triangles that are also
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 ...
s 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 ...
. However, 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 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'' ...
, there are infinitely many different shapes of right isosceles triangles.


30°–60°–90° triangle

This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° (), 60° (), and 90° (). The sides are in the ratio 1 :   : 2. The proof of this fact is clear using
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 ...
. The
geometric 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 ca ...
proof is: :Draw an equilateral triangle ''ABC'' with side length 2 and with point ''D'' as the midpoint of segment ''BC''. Draw an altitude line from ''A'' to ''D''. Then ''ABD'' is a 30°–60°–90° triangle with hypotenuse of length 2, and base ''BD'' of length 1. :The fact that the remaining leg ''AD'' has length follows immediately from 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 30°–60°–90° triangle is the only right triangle whose angles are in an
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
. The proof of this fact is simple and follows on from the fact that if ''α'', , are the angles in the progression then the sum of the angles = 180°. After dividing by 3, the angle must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.


Side-based

Right triangles whose sides are of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
lengths, with the sides collectively known as
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 ...
s, possess angles that cannot all be
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
of degrees. (This follows from
Niven's theorem In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of ''θ'' in the interval 0° ≤ ''θ'' ≤ 90° for which the sine of ''θ'' degrees is also a rational number ...
.) They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio : where ''m'' and ''n'' are any positive integers such that .


Common Pythagorean triples

There are several Pythagorean triples which are well-known, including those with sides in the ratios: : The 3 : 4 : 5 triangles are the only right triangles with edges in
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
. Triangles based on Pythagorean triples are Heronian, meaning they have integer
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
as well as integer sides. The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated. It was first conjectured by the historian
Moritz Cantor Moritz Benedikt Cantor (23 August 1829 – 10 April 1920) was a German historian of mathematics. Biography Cantor was born at Mannheim. He came from a Sephardi Jewish family that had emigrated to the Netherlands from Portugal Portugal, off ...
in 1882. It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement; that
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for his ''P ...
recorded in '' Isis and Osiris'' (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle; and that the
Berlin Papyrus 6619 The Berlin Papyrus 6619, simply called the Berlin Papyrus when the context makes it clear, is one of the primary sources of ancient Egyptian mathematics. One of the two mathematics problems on the Papyrus may suggest that the ancient Egyptians kn ...
from the
Middle Kingdom of Egypt The Middle Kingdom of Egypt (also known as The Period of Reunification) is the period in the history of ancient Egypt following a period of political division known as the First Intermediate Period. The Middle Kingdom lasted from approximatel ...
(before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other." The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem." Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain: he guesses that the Ancient Egyptians probably did know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles". The following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones in lowest form in the list above) with both non-hypotenuse sides less than 256: :


Almost-isosceles Pythagorean triples

Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is and cannot be expressed as a ratio of two integers. However, infinitely many ''almost-isosceles'' right triangles do exist. These are right-angled triangles with integer sides for which the lengths of the non-hypotenuse edges differ by one. Such almost-isosceles right-angled triangles can be obtained recursively, :''a''0 = 1, ''b''0 = 2 :''a''''n'' = 2''b''''n''−1 + ''a''''n''−1'' :''b''''n'' = 2''a''''n'' + ''b''''n''−1'' ''a''''n'' is length of hypotenuse, ''n'' = 1, 2, 3, .... Equivalently, :(\tfrac)^2+(\tfrac)^2 = y^2 where are solutions to the
Pell equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...
, with the hypotenuse ''y'' being the odd terms of the
Pell numbers In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , and ...
1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... .. The smallest Pythagorean triples resulting are: : Alternatively, the same triangles can be derived from the
square triangular number In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are: :0, 1, 36, , , , , , , Expl ...
s.


Arithmetic and geometric progressions

The Kepler triangle is a right triangle whose sides are in
geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...
. If the sides are formed from the geometric progression ''a'', ''ar'', ''ar''2 then its common ratio ''r'' is given by ''r'' = where ''φ'' is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
. Its sides are therefore in the ratio . Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that its sides be in geometric progression. The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
.


Sides of regular polygons

Let a=2\sin\frac=\frac=\frac1\varphi\approx 0.618 be the side length of a regular
decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' i ...
inscribed in the unit circle, where \varphi is the golden ratio. Let b=2\sin\frac=1 be the side length of a regular
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 ...
in the unit circle, and let c=2\sin\frac=\sqrt\approx 1.176 be the side length of a regular
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
in the unit circle. Then a^2+b^2=c^2, so these three lengths form the sides of a right triangle. The same triangle forms half of a
golden rectangle In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618. Golden rectangles exhibit a special form of self-similarity ...
. It may also be found within a
regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...
of side length c: the shortest line segment from any vertex V to the plane of its five neighbors has length a, and the endpoints of this line segment together with any of the neighbors of V form the vertices of a right triangle with sides a, b, and c.nLab: pentagon decagon hexagon identity


See also

*
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 ...
*
Spiral of Theodorus In geometry, the spiral of Theodorus (also called ''square root spiral'', ''Einstein spiral'', ''Pythagorean spiral'', or ''Pythagoras's snail'') is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene ...


References


External links


3 : 4 : 5 triangle




with interactive animations {{DEFAULTSORT:Special Right Triangles Euclidean plane geometry Types of triangles