Pythagorean Theorem
The pythagorean theorem equates the square of the sides of a right triangle together. It is used throughout mathmatics. For example, the equation can be used to find the unknown length of a right triangle, verify that a triangle is a right triangle and more.
This example demonstrates the pythagorean theorem for a right triangle with sides length , , and . The pythagorean theorem equates the squares of the sides of the right triangle together.

Set up pythagorean theorem and substitute sides of the right triangle.

Evaulate the exponent expressions.

Add the numbers on the left side. Visually this can be respresented as the area of the side squares equals the area of the hypotenuse square.

The theorem holds for a right triangle with sides 3, 4, and 5.
45 45 90 Triangle
For example, if we have a right triangle where the length of the adjacent and opposite side are 1, we can use the theorem to find the length of the hypotenuse.
30 60 90 Triangle
In this example, we can use the theorem to find the length of the adjacent side of the following right triangle.
The pythagorean identity relates the sides of the right triangle together using only the angle of the right triangle. The identity is derived using pythagorean's theorem and the properties of the unit circle.