thumb|320px|Position of some special triangles in an [[Euler diagram of types of triangles, using the definition that isosceles triangles have at least two equal sides, i.e. that equilateral triangles are isosceles]]

A special right triangle is a right triangle with some notable feature that makes calculations on the triangle easier, or for which simple formulas exist.

The various relationships between the angles and sides of such triangles allow one to quickly calculate some useful quantities in geometric problems without resorting to more advanced methods.

Angle-based

right|thumb|Special angle-based triangles inscribed in a unit circle are handy for visualizing and remembering [[trigonometric functions of multiples of 30 and 45 degrees.]]

Angle-based special right triangles are those involving some special relationship between the triangle's three angle measures. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or radians, is equal to the sum of the other two angles.

The side lengths of these triangles can be deduced based on the unit circle, or with the use of other geometric methods; and these approaches may be extended to produce the values of trigonometric functions for some common angles, shown in the table below.

{| class="wikitable"

! degrees !! radians !! gons !! turns !! sin !! cos !! tan !! cotan

|-

| 0° || 0 || 0<sup>g</sup> || 0 || = 0 || = 1 || 0 || undefined

|-

| 30° || || <sup>g</sup> || || = || || ||

|-

| 45° || || 50<sup>g</sup> || || = || = || 1 || 1

|-

| 60° || || <sup>g</sup> || || || = || ||

|-

| 90° || || 100<sup>g</sup> || || = 1 || = 0 || undefined || 0

|}

The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the equilateral/equiangular (60°–60°–60°) triangle are the three Möbius triangles in the plane, meaning that they tessellate the plane via reflections in their sides; see Triangle group.

45°–45°–90° triangle

thumb|x150px|left|[[Set square shaped as 45°–45°–90° triangle]]

thumb|right|x150px|The side lengths of a 45°–45°–90° triangle

thumb|45°–45°–90° [[right triangle of hypotenuse length 1]]

In plane geometry, dividing a square along its diagonal results in two isosceles right triangles, each with one right angle (90°, radians) and two other congruent angles each measuring half of a right angle (45°, or radians). The sides in this triangle are in the ratio 1&nbsp;:&nbsp;1&nbsp;:&nbsp;, which follows immediately from the Pythagorean theorem.

Of all right triangles, such 45°–45°–90° degree triangles have the smallest ratio of the hypotenuse to the sum of the legs, namely . and the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely . (This follows from Niven's theorem.) 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:

:{| border="0" cellpadding="1" cellspacing="0"

|align="right"|3 :||align="right"| 4 :||align="right"| 5

|-

|align="right"|5 :||align="right"|12 :||align="right"|13

|-

|align="right"|8 :||align="right"|15 :||align="right"|17

|-

|align="right"|7 :||align="right"|24 :||align="right"|25

|-

|align="right"|9 :||align="right"|40 :||align="right"|41

|}

The 3&nbsp;:&nbsp;4&nbsp;:&nbsp;5 triangles are the only right triangles with edges in arithmetic progression. Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides.

The possible use of the 3&nbsp;:&nbsp;4&nbsp;:&nbsp;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 in 1882. 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." Such almost-isosceles right-angled triangles can be obtained recursively,

:a<sub>0</sub> = 1, b<sub>0</sub> = 2

:a<sub>n</sub> = 2b<sub>n−1</sub> + a<sub>n−1</sub>

:b<sub>n</sub> = 2a<sub>n</sub> + b<sub>n−1</sub>

a<sub>n</sub> is length of hypotenuse, n = 1, 2, 3, .... Equivalently,

:<math>(\tfrac{x-1}{2})^2+(\tfrac{x+1}{2})^2 = y^2</math>

where {x, y} are solutions to the Pell equation , with the hypotenuse y being the odd terms of the Pell numbers 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... .. The smallest Pythagorean triples resulting are:

:{| border="0" cellpadding="1" cellspacing="0" align="left" style="margin-right: 3em"

|align="right"| 3 :||align="right"| 4 :||align="right"| 5

|-

|align="right"| 20 :||align="right"| 21 :||align="right"| 29

|-

|align="right"| 119 :||align="right"| 120 :||align="right"| 169

|-

|align="right"| 696 :||align="right"| 697 :||align="right"| 985

|}

{| border="0" cellpadding="1" cellspacing="0" align="left"

|align="right"| 4,059 :||align="right"| 4,060 :||align="right"| 5,741

|-

|align="right"| 23,660 :||align="right"| 23,661 :||align="right"| 33,461

|-

|align="right"| 137,903 :||align="right"| 137,904 :||align="right"| 195,025

|-

|align="right"| 803,760 :||align="right"| 803,761 :||align="right"| 1,136,689

|}

Alternatively, the same triangles can be derived from the square triangular numbers.

Arithmetic and geometric progressions

right|thumb|A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the [[golden ratio.]]

The Kepler triangle is a right triangle whose sides are in geometric progression. If the sides are formed from the geometric progression a, ar, ar<sup>2</sup> then its common ratio r is given by r = where φ is the golden ratio. 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.

Sides of regular polygons

thumb|The sides of a pentagon, hexagon, and decagon, inscribed in [[Congruence (geometry)|congruent circles, form a right triangle.]]

Let <math display=block>a=2\sin\frac{\pi}{10}=\frac{-1+\sqrt5}{2}=\frac1\varphi\approx 0.618</math> be the side length of a regular decagon inscribed in the unit circle, where <math>\varphi</math> is the golden ratio. Let <math display=block>b=2\sin\frac{\pi}{6}=1</math> be the side length of a regular hexagon in the unit circle, and let <math display=block>c=2\sin\frac{\pi}{5}=\sqrt{\frac{5-\sqrt5}{2\approx 1.176</math> be the side length of a regular pentagon in the unit circle. Then <math>a^2+b^2=c^2</math>, so these three lengths form the sides of a right triangle. The same triangle forms half of a golden rectangle. It may also be found within a regular icosahedron of side length <math>c</math>: the shortest line segment from any vertex <math>V</math> to the plane of its five neighbors has length <math>a</math>, and the endpoints of this line segment together with any of the neighbors of <math>V</math> form the vertices of a right triangle with sides <math>a</math>, <math>b</math>, and <math>c</math>.

See also

  • Ailles rectangle, combining several special right triangles
  • Integer triangle
  • Spiral of Theodorus

References