Overview
A right triangle is a plane triangle that contains one angle equal to 90 degrees. Because the angles of any triangle sum to 180 degrees, the remaining two angles in a right triangle are complementary and together measure 90 degrees. You will also see this shape called a right-angled triangle in some texts.
Parts and basic properties
The side opposite the 90° angle is the hypotenuse, typically the longest side of the triangle. The other two sides are called the legs or catheti. The 90° measure itself is an angle, and each side may be referred to generically as a side in geometric descriptions. Important simple facts include that the hypotenuse is always opposite the right angle and that the two legs meet at the right angle.
Fundamental theorem and trigonometry
The relationship between the lengths of the sides is governed by the Pythagorean theorem: in a right triangle the sum of the squares of the legs equals the square of the hypotenuse. To find unknown angles or side lengths many problems use elementary trigonometry—ratios such as sine, cosine and tangent that relate an acute angle to the lengths of the opposite, adjacent and hypotenuse sides.
Uses, problem solving and examples
Right triangles appear across mathematics, physics, engineering and surveying. They serve as the basis for distance calculations, slope and gradient problems, and resolving vectors into components. Practical tasks—measuring height indirectly, designing right-angled supports, or computing line-of-sight distances—rely on solving right triangles with theorems and trig ratios.
Notable facts and distinctions
- Any triangle with one 90° angle is a right triangle; non-right triangles are acute or obtuse depending on their largest angle.
- Special right triangles (for example 45°–45°–90° and 30°–60°–90°) have fixed side ratios often used for quick calculations.
- Historical development of right-triangle results predates classical attributions and appears in several ancient mathematical traditions.
For introductory definitions, constructions, and worked examples consult general geometry resources: see a basic entry on the hypotenuse and further materials on angle measures and solution techniques at linked references (sides, angles).