Special right triangle

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 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 ⁠π/2⁠ 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.

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.

In plane geometry, dividing a 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.

Of all right triangles, such 45°–45°–90° degree triangles have the smallest ratio of the hypotenuse to the sum of the legs, namely ⁠√2/2⁠. and the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely ⁠√2/4⁠.

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.