# Special right triangle

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Right triangle with a feature making calculations on the triangle easier

"45-45-90 triangle" and "30-60-90 triangle" redirect here. For the drawing tool, see [Set square](/source/Set_square).

Position of some special triangles in an [Euler diagram](/source/Euler_diagram) of types of triangles, using the definition that [isosceles triangles](/source/Isosceles_triangle) have at least two equal sides, i.e. that [equilateral triangles](/source/Equilateral_triangle) are isosceles

A **special right triangle** is a [right triangle](/source/Right_triangle) with some notable feature that makes calculations on the [triangle](/source/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](/source/Geometry) 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](/source/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](/source/Degree_(angle)) or ⁠π/2⁠ [radians](/source/Radian), is equal to the sum of the other two angles.

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

degrees radians gons turns sin cos tan cotan 0° 0 0g 0 ⁠√0/2⁠ = 0 ⁠√4/2⁠ = 1 0 undefined 30° ⁠π/6⁠ ⁠33+1/3⁠g ⁠1/12⁠ ⁠√1/2⁠ = ⁠1/2⁠ ⁠√3/2⁠ ⁠1/√3⁠ √3 45° ⁠π/4⁠ 50g ⁠1/8⁠ ⁠√2/2⁠ = ⁠1/√2⁠ ⁠√2/2⁠ = ⁠1/√2⁠ 1 1 60° ⁠π/3⁠ ⁠66+2/3⁠g ⁠1/6⁠ ⁠√3/2⁠ ⁠√1/2⁠ = ⁠1/2⁠ √3 ⁠1/√3⁠ 90° ⁠π/2⁠ 100g ⁠1/4⁠ ⁠√4/2⁠ = 1 ⁠√0/2⁠ = 0 undefined 0

45°–45°–90°

30°–60°–90°

The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the [equilateral](/source/Equilateral_triangle)/equiangular (60°–60°–60°) triangle are the three [Möbius triangles](/source/M%C3%B6bius_triangle) in the plane, meaning that they [tessellate](/source/Tessellate) the plane via [reflections](/source/Reflection_(mathematics)) in their sides; see [Triangle group](/source/Triangle_group).

### 45°–45°–90° triangle

[Set square](/source/Set_square) shaped as 45°–45°–90° triangle

The side lengths of a 45°–45°–90° triangle

45°–45°–90° [right triangle](/source/Right_triangle) of [hypotenuse](/source/Hypotenuse) length 1

In [plane geometry](/source/Plane_geometry), dividing a [square](/source/Square) along its diagonal results in two **isosceles right triangles**, each with one right angle (90°, ⁠π/2⁠ radians) and two other congruent angles each measuring half of a right angle (45°, or ⁠π/4⁠ radians). The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from the [Pythagorean theorem](/source/Pythagorean_theorem).

Of all right triangles, such 45°–45°–90° degree triangles have the smallest ratio of the [hypotenuse](/source/Hypotenuse) to the sum of the legs, namely ⁠√2/2⁠.[1]: p. 282, p. 358 and the greatest ratio of the [altitude](/source/Altitude_(triangle)) 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](/source/Isosceles_triangle) in [Euclidean geometry](/source/Euclidean_geometry). However, in [spherical geometry](/source/Spherical_geometry) and [hyperbolic geometry](/source/Hyperbolic_geometry), there are infinitely many different shapes of right isosceles triangles.

### 30°–60°–90° triangle

Set square, shaped as 30°–60°–90° triangle

The side lengths of a 30°–60°–90° triangle

30°–60°–90° [right triangle](/source/Right_triangle) of [hypotenuse](/source/Hypotenuse) length 1

Another type of special right triangle is the 30°-60°-90° triangle, which refers to any triangle with those three angle measures. Notably, these angles are in the ratio 1 : 2 : 3.

A useful property of such triangles is that their side lengths are in the ratio 1 : [√3](/source/Square_root_of_3) : 2. This property can be proven using [trigonometry](/source/Trigonometry), or via the [geometric](/source/Geometry) proof below:

Draw an equilateral triangle *ABC* with side length 2, and with point *M* as the midpoint of segment *BC*. Draw an altitude line from *A* to *M*. Then *ABM* is a 30°–60°–90° triangle with hypotenuse of length 2, and base *BM* of length 1.

The fact that the remaining leg *AM* has length [√3](/source/Square_root_of_3) follows immediately from the [Pythagorean theorem](/source/Pythagorean_theorem).

The 30°–60°–90° triangle is the only right triangle whose angles are in an [arithmetic progression](/source/Arithmetic_progression). The proof of this fact is simple and follows on from the fact that if *α*, *α* + *δ*, *α* + 2*δ* are the angles in the progression then the sum of the angles 3*α* + 3*δ* = 180°. After dividing by 3, the angle *α* + *δ* must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.

## Side-based

Right triangles whose sides are of [integer](/source/Integer) lengths, with the sides collectively known as [Pythagorean triples](/source/Pythagorean_triple), possess angles that cannot all be [rational numbers](/source/Rational_numbers) of [degrees](/source/Degree_(angle)).[2] (This follows from [Niven's theorem](/source/Niven's_theorem).) They are most useful in that they may be easily remembered and any [multiple](/source/Multiple_(mathematics)) 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 : 2*mn* : *m*2 + *n*2

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

### Common Pythagorean triples

Main article: [Pythagorean triple](/source/Pythagorean_triple)

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](/source/Arithmetic_progression). Triangles based on Pythagorean triples are [Heronian](/source/Heronian_triangle), meaning they have integer [area](/source/Area) as well as integer sides.

The possible use of the 3 : 4 : 5 triangle in [Ancient Egypt](/source/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.[3] It was first conjectured by the historian [Moritz Cantor](/source/Moritz_Cantor) in 1882.[3] It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement;[3] that [Plutarch](/source/Plutarch) recorded in *[Isis and Osiris](/source/De_Iside_et_Osiride)* (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle;[3] and that the [Berlin Papyrus 6619](/source/Berlin_Papyrus_6619) from the [Middle Kingdom of Egypt](/source/Middle_Kingdom_of_Egypt) (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 ⁠1/2⁠ + ⁠1/4⁠ the side of the other."[4] 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."[3] 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".[3]

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:

- 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, because the ratio of the hypotenuse to either other side is √2 and [√2 cannot be expressed as a ratio of two integers](/source/Square_root_of_2#Proofs_of_irrationality). 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](/source/Cathetus) differ by one.[5][6] 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,

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

where {*x*, *y*} are solutions to the [Pell equation](/source/Pell_equation) *x*2 − 2*y*2 = −1, with the hypotenuse *y* being the odd terms of the [Pell numbers](/source/Pell_numbers) **1**, 2, **5**, 12, **29**, 70, **169**, 408, **985**, 2378... (sequence [A000129](https://oeis.org/A000129) in the [OEIS](/source/On-Line_Encyclopedia_of_Integer_Sequences)).. The smallest Pythagorean triples resulting are:[7]

- 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

Alternatively, the same triangles can be derived from the [square triangular numbers](/source/Square_triangular_number).[8]

### Arithmetic and geometric progressions

A **Kepler triangle** is a right triangle formed by three squares with areas in geometric progression according to the **[golden ratio](/source/Golden_ratio)**.

Main article: [Kepler triangle](/source/Kepler_triangle)

The Kepler triangle is a right triangle whose sides are in [geometric progression](/source/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](/source/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 geometric progression.

The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in [arithmetic progression](/source/Arithmetic_progression).[9]

### Sides of regular polygons

The sides of a pentagon, hexagon, and decagon, inscribed in [congruent](/source/Congruence_(geometry)) circles, form a right triangle.

Let a = 2 sin ⁡ π 10 = − 1 + 5 2 = 1 φ ≈ 0.618 {\displaystyle a=2\sin {\frac {\pi }{10}}={\frac {-1+{\sqrt {5}}}{2}}={\frac {1}{\varphi }}\approx 0.618} be the side length of a regular [decagon](/source/Decagon) inscribed in the unit circle, where φ {\displaystyle \varphi } is the golden ratio. Let b = 2 sin ⁡ π 6 = 1 {\displaystyle b=2\sin {\frac {\pi }{6}}=1} be the side length of a regular [hexagon](/source/Hexagon) in the unit circle, and let c = 2 sin ⁡ π 5 = 5 − 5 2 ≈ 1.176 {\displaystyle c=2\sin {\frac {\pi }{5}}={\sqrt {\frac {5-{\sqrt {5}}}{2}}}\approx 1.176} be the side length of a regular [pentagon](/source/Pentagon) in the unit circle. Then a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} , so these three lengths form the sides of a right triangle.[10] The same triangle forms half of a [golden rectangle](/source/Golden_rectangle). It may also be found within a [regular icosahedron](/source/Regular_icosahedron) of side length c {\displaystyle c} : the shortest line segment from any vertex V {\displaystyle V} to the plane of its five neighbors has length a {\displaystyle a} , and the endpoints of this line segment together with any of the neighbors of V {\displaystyle V} form the vertices of a right triangle with sides a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} .[11]

## See also

- [Ailles rectangle](/source/Ailles_rectangle), combining several special right triangles

- [Integer triangle](/source/Integer_triangle)

- [Spiral of Theodorus](/source/Spiral_of_Theodorus)

## References

1. ^ [***a***](#cite_ref-PL_1-0) [***b***](#cite_ref-PL_1-1) Posamentier, Alfred S., and Lehman, Ingmar. *[The Secrets of Triangles](/source/The_Secrets_of_Triangles)*. Prometheus Books, 2012.

1. **[^](#cite_ref-2)** Weisstein, Eric W. ["Rational Triangle"](http://mathworld.wolfram.com/RationalTriangle.html). *MathWorld*.

1. ^ [***a***](#cite_ref-Cooke2011_3-0) [***b***](#cite_ref-Cooke2011_3-1) [***c***](#cite_ref-Cooke2011_3-2) [***d***](#cite_ref-Cooke2011_3-3) [***e***](#cite_ref-Cooke2011_3-4) [***f***](#cite_ref-Cooke2011_3-5) Cooke, Roger L. (2011). [*The History of Mathematics: A Brief Course*](https://books.google.com/books?id=wOGh7XPowAMC) (2nd ed.). John Wiley & Sons. pp. 237–238. [ISBN](/source/ISBN_(identifier)) [978-1-118-03024-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-118-03024-0).

1. **[^](#cite_ref-4)** Gillings, Richard J. (1982). [*Mathematics in the Time of the Pharaohs*](https://archive.org/details/mathematicsintim0000gill). Dover. p. [161](https://archive.org/details/mathematicsintim0000gill/page/161).

1. **[^](#cite_ref-5)** Forget, T. W.; Larkin, T. A. (1968), ["Pythagorean triads of the form *x*, *x* + 1, *z* described by recurrence sequences"](http://www.fq.math.ca/Scanned/6-3/6-3/forget.pdf) (PDF), *Fibonacci Quarterly*, **6** (3): 94–104, [doi](/source/Doi_(identifier)):[10.1080/00150517.1968.12431232](https://doi.org/10.1080%2F00150517.1968.12431232).

1. **[^](#cite_ref-6)** Chen, C. C.; Peng, T. A. (1995), ["Almost-isosceles right-angled triangles"](http://ajc.maths.uq.edu.au/pdf/11/ajc-v11-p263.pdf) (PDF), *The Australasian Journal of Combinatorics*, **11**: 263–267, [MR](/source/MR_(identifier)) [1327342](https://mathscinet.ams.org/mathscinet-getitem?mr=1327342).

1. **[^](#cite_ref-7)** (sequence [A001652](https://oeis.org/A001652) in the [OEIS](/source/On-Line_Encyclopedia_of_Integer_Sequences))

1. **[^](#cite_ref-8)** Nyblom, M. A. (1998), ["A note on the set of almost-isosceles right-angled triangles"](http://www.fq.math.ca/Scanned/36-4/nyblom.pdf) (PDF), *The Fibonacci Quarterly*, **36** (4): 319–322, [doi](/source/Doi_(identifier)):[10.1080/00150517.1998.12428915](https://doi.org/10.1080%2F00150517.1998.12428915), [MR](/source/MR_(identifier)) [1640364](https://mathscinet.ams.org/mathscinet-getitem?mr=1640364).

1. **[^](#cite_ref-9)** Beauregard, Raymond A.; Suryanarayan, E. R. (1997), "Arithmetic triangles", *[Mathematics Magazine](/source/Mathematics_Magazine)*, **70** (2): 105–115, [doi](/source/Doi_(identifier)):[10.2307/2691431](https://doi.org/10.2307%2F2691431), [JSTOR](/source/JSTOR_(identifier)) [2691431](https://www.jstor.org/stable/2691431), [MR](/source/MR_(identifier)) [1448883](https://mathscinet.ams.org/mathscinet-getitem?mr=1448883).

1. **[^](#cite_ref-10)** [Euclid's *Elements*, Book XIII, Proposition 10](http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII10.html).

1. **[^](#cite_ref-11)** [nLab: pentagon decagon hexagon identity](http://ncatlab.org/nlab/show/pentagon+decagon+hexagon+identity).

## External links

- [3 : 4 : 5 triangle](https://www.mathopenref.com/triangle345.html)

- [30–60–90 triangle](https://www.mathopenref.com/triangle306090.html)

- [45–45–90 triangle](https://www.mathopenref.com/triangle454590.html) – with interactive animations

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Adapted from the Wikipedia article [Special right triangle](https://en.wikipedia.org/wiki/Special_right_triangle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Special_right_triangle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.