Right Triangle and Pythagorean Calculator

How do you solve a right triangle?

A right triangle has one 90 degree angle. The longest side, opposite the right angle, is the hypotenuse, and the other two sides are the legs. Two pieces of information are enough to solve it: two sides, or one side and one acute angle.

The Pythagorean theorem

The squares of the legs add up to the square of the hypotenuse: c² = a² + b². So c = √(a² + b²) and a = √(c² − b²).

Angles

The acute angles follow from side ratios: sin α = a / c, cos α = b / c, tan α = a / b. The two acute angles add up to 90 degrees.

Example

With legs of 3 m and 4 m the hypotenuse is √(9 + 16) = 5 m. The area is 3 × 4 / 2 = 6 m², the perimeter 12 m and the altitude to the hypotenuse 3 × 4 / 5 = 2.4 m. The angles are about 36.87 and 53.13 degrees.

Use on site

The 3 4 5 rule checks whether a corner is square: mark 3 m along one side and 4 m along the other, and the corner is square if the marks are 5 m apart. Larger layouts use multiples of the same ratio (6 8 10, 9 12 15). Roof pitch, ramp length and stair calculations are right triangle problems as well.

Limits

The calculator handles right triangles only. The hypotenuse is always longer than either leg, and the acute angle must lie between 0 and 90 degrees.