Special right triangle
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 "anglebased" right triangle. A "sidebased" 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.
Contents
Anglebased
"Anglebased" 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:
Degrees  Radians  Gons  Turns  sin  cos  tan  cotan 

0°  0  0^{g}  0  √0/2 = 0  √4/2 = 1  0  undefined 
30°  π/6  33 1/3^{g}  1/12  √1/2 = 1/2  √3/2  1/√3  √3 
45°  π/4  50^{g}  1/8  √2/2 = 1/√2  √2/2 = 1/√2  1  1 
60°  π/3  66 2/3^{g}  1/6  √3/2  √1/2 = 1/2  √3  1/√3 
90°  π/2  100^{g}  1/4  √4/2 = 1  √0/2 = 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
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.
Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely √2/2.^{[1]}^{:p.282,p.358}
Of all right triangles, the 45°–45°–90° degree triangle has the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely √2/4.^{[1]}^{:p.282}
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
This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° (π/6), 60° (π/3), and 90° (π/2). 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°. After dividing by 3, the angle α + δ must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.
Sidebased
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 wellknown, 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 in lowest form in the list above) with both nonhypotenuse 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 
Almostisosceles Pythagorean triples
Isosceles rightangled triangles cannot have sides with integer values, because the ratio of the hypotenuse to a either other side is √2, but √2 cannot be expressed as a ratio of two integers. However, infinitely many almostisosceles right triangles do exist. These are rightangled triangles with integral sides for which the lengths of the nonhypotenuse edges differ by one.^{[3]}^{[4]} Such almostisosceles rightangled triangles can be obtained recursively,
 a_{0} = 1, b_{0} = 2
 a_{n} = 2b_{n−1} + a_{n−1}
 b_{n} = 2a_{n} + b_{n−1}
a_{n} is length of hypotenuse, n = 1, 2, 3, .... Equivalently,
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 4,059 : 4,060 : 5,741 23,660 : 23,661 : 33,461 137,903 : 137,904 : 195,025 803,760 : 803,761 : 1,136,689 4,684,659 : 4,684,660 : 6,625,109
Alternatively, the same triangles can be derived from the square triangular numbers.^{[6]}
Arithmetic and geometric progressions
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, ar^{2} then its common ratio r is given by r = √φ where φ is the golden ratio. Its sides are therefore in the ratio 1 : √φ : φ. Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that its sides be in a geometric progression.
The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in an arithmetic progression.^{[7]}
Sides of regular polygons
Let a = 2 sin π/10 = −1 + √5/2 = 1/φ be the side length of a regular decagon inscribed in the unit circle, where φ is the golden ratio. Let b = 2 sin π/6 = 1 be the side length of a regular hexagon in the unit circle, and let c = 2 sin π/5 = be the side length of a regular pentagon in the unit circle. Then a^{2} + b^{2} = c^{2}, so these three lengths form the sides of a right triangle.^{[8]} The same triangle forms half of a golden rectangle. It may also be found within a regular icosahedron 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.^{[9]}
See also
References
 ↑ ^{1.0} ^{1.1} Posamentier, Alfred S., and Lehman, Ingmar. The Secrets of Triangles. Prometheus Books, 2012.
 ↑ Weisstein, Eric W. "Rational Triangle". MathWorld.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Forget, T. W.; Larkin, T. A. (1968), "Pythagorean triads of the form x, x + 1, z described by recurrence sequences" (PDF), Fibonacci Quarterly, 6 (3): 94–104<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ Chen, C. C.; Peng, T. A. (1995), "Almostisosceles rightangled triangles" (PDF), The Australasian Journal of Combinatorics, 11: 263–267, MR 1327342<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ (sequence A001652 in OEIS)
 ↑ Nyblom, M. A. (1998), "A note on the set of almostisosceles rightangled triangles" (PDF), The Fibonacci Quarterly, 36 (4): 319–322, MR 1640364<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ Beauregard, Raymond A.; Suryanarayan, E. R. (1997), "Arithmetic triangles", Mathematics Magazine, 70 (2): 105–115, doi:10.2307/2691431, MR 1448883<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
 ↑ Euclid's Elements, Book XIII, Proposition 10.
 ↑ nLab: pentagon decagon hexagon identity.
External links
 3 : 4 : 5 triangle
 30–60–90 triangle
 45–45–90 triangle With interactive animations