righttriangle

A right triangle, also called a right-angled triangle, is a triangle in which one angle measures 90 degrees. The two sides that meet at the right angle are called the legs, while the side opposite the right angle is the hypotenuse.

If the legs have lengths a and b and the hypotenuse has length c, then the Pythagorean theorem states a^2 + b^2 = c^2.

The area is A = (ab)/2 and the perimeter is P = a + b + c. The inradius r equals (a + b − c)/2 (also r = ab/(a + b + c)); the circumradius is R = c/2.

The other two angles are acute and sum to 90 degrees. For the angle A opposite side a, sin A = a/c, cos A = b/c, and tan A = a/b.

A common special case is the isosceles right triangle, where a = b; then c = a√2 and the angles are 45°, 45°, 90°.

Integer right triangles form Pythagorean triples. Primitive triples satisfy gcd(a, b, c) = 1; they can be generated by a = m^2 − n^2, b = 2mn, c = m^2 + n^2 with integers m > n, opposite parity, and gcd(m, n) = 1.

Right triangles appear in various fields, including surveying, architecture, and computer graphics. They are fundamental to trigonometry, as the definitions of sine, cosine, and tangent rely on right-triangle ratios.