Right Triangle Calculator (Pythagorean Theorem)

A right triangle has one 90° angle, so two values are enough to find everything else. Enter any two of the legs a and b, the hypotenuse c and the acute angles α and β, and this calculator finds the missing sides and angles with the Pythagorean theorem and basic trigonometry, plus the area, perimeter, the altitude to the hypotenuse, the inradius and the sine, cosine and tangent of α. With whole-number sides it also gives the exact answer in simplest radical form.

How to use this tool

1. Enter two known values — two sides, or one side and one acute angle.
2. Leave the other three boxes empty.
3. Read the missing values and the worked substitution.
4. Use the exact form (such as 2√5) when your homework asks for simplest radical form.

Formula

a² + b² = c² · sin α = a ÷ c · cos α = b ÷ c · tan α = a ÷ b · α + β = 90°

Leg a is opposite angle α and leg b is opposite β; c is always the longest side. The area is a × b ÷ 2, the altitude from the right angle to the hypotenuse is a × b ÷ c and the inradius is (a + b − c) ÷ 2. A square root is simplified by taking out square factors: √20 = √(4 × 5) = 2√5.

Example

With legs a = 3 and b = 4, c² = 9 + 16 = 25, so the hypotenuse is c = 5. Angle α = atan(3 ÷ 4) ≈ 36.87° and β ≈ 53.13°; the area is 6, the perimeter 12 and the altitude to the hypotenuse 3 × 4 ÷ 5 = 2.4.