Aspecial right triangle is aright triangle with some regular feature that makes calculations on thetriangle easier, or for which simple formulas exist. For example, a right triangle may haveangles 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 ofwhole numbers, such as 3 : 4 : 5, or of other special numbers such as thegolden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths ingeometric problems without resorting to more advanced methods.

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 90degrees or⁠π/2⁠radians, is equal to the sum of the other two angles.

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

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

45°–45°–90°

30°–60°–90°

The 45°–45°–90° triangle, the 30°–60°–90° triangle, and theequilateral/equiangular (60°–60°–60°) triangle are the threeMöbius triangles in the plane, meaning that theytessellate the plane viareflections in their sides; seeTriangle group.

Inplane geometry, dividing asquare along its diagonal results in twoisosceles right triangles, each with one right angle (90°,⁠π/2⁠ radians) and two other congruent angles each measuring half of a right angle (45°, or⁠π/4⁠ radians). The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from thePythagorean theorem.

Of all right triangles, such 45°–45°–90° degree triangles have the smallest ratio of thehypotenuse to the sum of the legs, namely⁠√2/2⁠.^{[1]: p. 282, p. 358} and the greatest ratio of thealtitude 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 alsoisosceles triangles inEuclidean geometry. However, inspherical geometry andhyperbolic geometry, there are infinitely many different shapes of right isosceles triangles.

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 usingtrigonometry. Thegeometric proof is:

Draw an equilateral triangleABC with side length 2 and with pointM as the midpoint of segmentBC. Draw an altitude line fromA toM. ThenABM is a 30°–60°–90° triangle with hypotenuse of length 2, and baseBM of length 1.

The fact that the remaining legAM has length√3 follows immediately from thePythagorean theorem.

The 30°–60°–90° triangle is the only right triangle whose angles are in anarithmetic progression. The proof of this fact is simple and follows on from the fact that ifα,α +δ,α + 2δ are the angles in the progression then the sum of the angles3α + 3δ = 180°. After dividing by 3, the angleα +δ must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.

Right triangles whose sides are ofinteger lengths, with the sides collectively known asPythagorean triples, possess angles that cannot all berational numbers ofdegrees.^[2] (This follows fromNiven's theorem.) They are most useful in that they may be easily remembered and anymultiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio

m² −n² : 2mn :m² +n²

wherem andn are any positive integers such thatm >n.

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 inarithmetic progression. Triangles based on Pythagorean triples areHeronian, meaning they have integerarea as well as integer sides.

The possible use of the 3 : 4 : 5 triangle inAncient 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 historianMoritz 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] thatPlutarch recorded inIsis and Osiris (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle;^[3] and that theBerlin Papyrus 6619 from theMiddle 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 is⁠1/2⁠ +⁠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

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. However, infinitely manyalmost-isosceles right triangles do exist. These are right-angled triangles with integer sides for which the lengths of thenon-hypotenuse edges differ by one.^[5][6] Such almost-isosceles right-angled triangles can be obtained recursively,

a₀ = 1,b₀ = 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,

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

where {x,y} are solutions to thePell equationx² − 2y² = −1, with the hypotenusey being the odd terms of thePell numbers1, 2,5, 12,29, 70,169, 408,985, 2378... (sequenceA000129 in theOEIS).. The smallest Pythagorean triples resulting are:^[7]

3 :	4 :	5
20 :	21 :	29
119 :	120 :	169
696 :	697 :	985

Alternatively, the same triangles can be derived from thesquare triangular numbers.^[8]

The Kepler triangle is a right triangle whose sides are ingeometric progression. If the sides are formed from the geometric progressiona,ar,ar² then its common ratior is given byr =√φ whereφ is thegolden ratio. Its sides are therefore in the ratio1 :√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 inarithmetic progression.^[9]

Let$${\displaystyle a=2\sin \frac{\pi }{10}=\frac{-1+\sqrt{5}}{2}=\frac{1}{\phi }\approx 0.618}$$ be the side length of a regulardecagon inscribed in the unit circle, where${\displaystyle \phi }$ is the golden ratio. Let$${\displaystyle b=2\sin \frac{\pi }{6}=1}$$ be the side length of a regularhexagon in the unit circle, and let$${\displaystyle c=2\sin \frac{\pi }{5}=\sqrt{\frac{5-\sqrt{5}}{2}}\approx 1.176}$$ be the side length of a regularpentagon in the unit circle. Then${\displaystyle a^2+b^2=c^2}$, so these three lengths form the sides of a right triangle.^[10] The same triangle forms half of agolden rectangle. It may also be found within aregular icosahedron of side length${\displaystyle c}$: the shortest line segment from any vertex${\displaystyle V}$ to the plane of its five neighbors has length${\displaystyle a}$, and the endpoints of this line segment together with any of the neighbors of${\displaystyle V}$ form the vertices of a right triangle with sides${\displaystyle a}$,${\displaystyle b}$, and${\displaystyle c}$.^[11]

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

• 3 : 4 : 5 triangle
• 30–60–90 triangle
• 45–45–90 triangle – with interactive animations

• Euclidean plane geometry
• Types of triangles

• Articles with short description
• Short description matches Wikidata