$Pythagorean Theorem$

Pythagorean Theorem

Find the hypotenuse from the legs or a leg from the hypotenuse, the area and angles of a right triangle, and check whether a triangle is right.

What is known:

Two legs — find the hypotenuse

Hypotenuse and leg — find the other leg

Three sides — check if the triangle is right

Leg a

Leg b

Units:

Lengths are positive numbers (whole or decimal), angles are in degrees.

Test yourself

The Pythagorean theorem calculator finds the hypotenuse from two legs or a leg from the hypotenuse and the other leg, as well as the area, perimeter, height to the hypotenuse and the acute angles of the right triangle. From three sides it checks whether the triangle is right.

The Pythagorean theorem

In a right triangle the square of the hypotenuse equals the sum of the squares of the legs:

c² = a² + b²

What to find	Formula
Hypotenuse	c = √(a² + b²)
Leg	b = √(c² − a²)
Height to the hypotenuse	h = ab / c
Area	S = ab / 2

Example 1. Legs 3 and 4.
c = √(9 + 16) = √25 = 5.

Example 2. Hypotenuse 13, leg 5.
b = √(169 − 25) = √144 = 12.

The converse

If the square of the longest side of a triangle equals the sum of the squares of the other two sides, the triangle is right. If it is greater, the triangle is obtuse; if it is less, the triangle is acute.

Pythagorean triples

Triples of natural numbers satisfying the theorem: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29). Multiplying a triple by any number gives another triple: (6, 8, 10), (9, 12, 15).