Pythagorean Theorem
The pythagorean theorem is a formula that equates the square of the sides of a right triangle together.
Examples
Pythagorean’s Theorem can be used to find an unknown length of a right triangle when given the length of two sides.
Pythagorean Theorem 3 4 5 Right Triangle
This example shows how the pythagorean theorem is true for a right triangle with sides length 3, 4, and 5. 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.
Usage
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.