Special right triangles

Position of some special triangles in an Euler diagram of types of triangles, using the definition that isosceles triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles.

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 45–45–90. 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. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.

Angle-based

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 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 π/2 radians, is equal to the sum of the other two angles.

The side lengths are generally deduced from the basis of the unit circle or other geometric methods. This approach may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°.

Special triangles are used to aid in calculating common trigonometric functions, as below:

0 0 $\tfrac{\sqrt{0}}{2}=0$ $\tfrac{\sqrt{4}}{2}=1$ $0$
30 $\tfrac{\pi}{6}$ $\tfrac{\sqrt{1}}{2}=\tfrac{1}{2}$ $\tfrac{\sqrt{3}}{2}$ $\tfrac{1}{\sqrt{3}}$
45 $\tfrac{\pi}{4}$ $\tfrac{\sqrt{2}}{2}=\tfrac{1}{\sqrt{2}}$ $\tfrac{\sqrt{2}}{2}=\tfrac{1}{\sqrt{2}}$ $1$
60 $\tfrac{\pi}{3}$ $\tfrac{\sqrt{3}}{2}$ $\tfrac{\sqrt{1}}{2}=\tfrac{1}{2}$ $\sqrt{3}$
90 $\tfrac{\pi}{2}$ $\tfrac{\sqrt{4}}{2}=1$ $\tfrac{\sqrt{0}}{2}=0$ $\infty$
45–45–90
30–60–90

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

The side lengths of a 45–45–90 triangle

In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or π radians. Hence, the angles respectively measure 45° (π/4), 45° (π/4), and 90° (π/2). The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from the Pythagorean theorem.

Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles.

30–60–90 triangle

The side lengths of a 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 : √3 : 2.

The proof of this fact is clear using trigonometry. The geometric 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 √3 follows immediately from the Pythagorean theorem.

The 30-60-90 triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α+δ, α+2δ are the angles in the progression then the sum of the angles 3α+3δ = 180°. So one angle must be 60° the other 90° leaving the remaining angle to be 30°.

Right triangle whose angles are in a geometric progression

The 30–60–90 triangle is the only right triangle whose angles are in an arithmetic progression. There is also a unique right triangle whose angles are in a geometric progression. The three angles are π/(2φ2), π/(2φ), π/2 where the common ratio is φ, the golden ratio.[1] Consequently the angles are in the ratio $1:\varphi:\varphi^2.\,$

Based on the sine rule, the sides are in the ratio $\sin{\frac{\pi}{2\varphi^2}}:\sin{\frac{\pi}{2\varphi}}:1.\,$ Because the sides are subject to the Pythagorean theorem, then $\sin^2{\frac{\pi}{2\varphi^2}}+\sin^2{\frac{\pi}{2\varphi}}=1\,$ and this resolves to the identity[citation needed]

$\cos{\frac{\pi}{\varphi+1}}+\cos{\frac{\pi}{\varphi}}=0.$

Interestingly by using the exponential definition of cosine, this can now be expanded into a phi identity that uses φ and the five fundamental mathematical constants π, e, i, 1, 0 of Euler's identity (though not as elegantly as the latter) as follows:

$e^{\frac{i\pi}{\varphi+1}}+e^{-\frac{i\pi}{\varphi+1}}+e^{\frac{i\pi}{\varphi}}+e^{-\frac{i\pi}{\varphi}}=0.\,$

Side-based

Right triangles whose sides are of integer lengths, Pythagorean triples, possess angles that cannot all be rational numbers of degrees.[2] 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

$m^2-n^2 : 2mn : m^2+n^2\,$

where m and n are any positive integers such that m>n.

Common Pythagorean triples

There are several Pythagorean triples which are well-known, including those with sides in the ratios:

3: 4 :5
5: 12 :13
8: 15 :17
7: 24 :25
9: 40 :41

The 3 : 4 : 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 following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones, listed above) with both non-hypotenuse sides less than 256:

11: 60 :61
12: 35 :37
13: 84 :85
15: 112 :113
16: 63 :65
17: 144 :145
19: 180 :181
20: 21 :29
20: 99 :101
21: 220 :221
24: 143 :145
28: 45 :53
28: 195 :197
32: 255 :257
33: 56 :65
36: 77 :85
39: 80 :89
44: 117 :125
48: 55 :73
51: 140 :149
52: 165 :173
57: 176 :185
60: 91 :109
60: 221 :229
65: 72 :97
84: 187 :205
85: 132 :157
88: 105 :137
95: 168 :193
96: 247 :265
104: 153 :185
105: 208 :233
115: 252 :277
119: 120 :169
120: 209 :241
133: 156 :205
140: 171 :221
160: 231 :281
161: 240 :289
204: 253 :325
207: 224 :305

Almost-isosceles Pythagorean triples

Isosceles right-angled triangles cannot have sides with integer values. However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integral sides for which the lengths of the non-hypotenuse edges differ by one.[3][4] Such almost-isosceles right-angled triangles can be obtained recursively,

a0 = 1, b0 = 2
an = 2bn–1 + an–1
bn = 2an + bn–1

an is length of hypotenuse, n = 1, 2, 3, .... Equivalently,

$(\tfrac{x-1}{2})^2+(\tfrac{x+1}{2})^2 = y^2$

where {x, y} are the solutions to the Pell equation $x^2-2y^2 = -1$, with the hypotenuse y being the odd terms of the Pell numbers 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... (sequence A000129 in OEIS).. The smallest Pythagorean triples resulting are:[5]

3 : 4  : 5
20 : 21  : 29
119  : 120  : 169
696  : 697  : 985
4059  : 4060  : 5741
23660  : 23661  : 33461

Alternatively, the same triangles can be derived from the square triangular numbers.[6]

Right triangle whose sides are in a geometric progression

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 a geometric progression. If the sides are formed from the geometric progression a, ar, ar2 then its common ratio r is given by r = √φ where φ is the golden ratio. Its sides are therefore in the ratio $1:\sqrt{\varphi}:\varphi .\,$