# What Is the Pythagorean Theorem Formula?

The legs and the hypotenuse in a right triangle are connected through the Pythagorean theorem.

Formula

### PythagoreanTheorem

The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs:

 ${a}^{2}+{b}^{2}={c}^{2}$

Rule

### TheTwoUsesofthePythagoreanTheorem

The Pythagorean theorem is mainly used for two purposes:

1.
When you have a right triangle, you are going to find the length of one of the sides of the triangle, and you know the lengths of the other two sides.
2.
When you have an arbitrary triangle, and you want to determine whether it’s a right triangle by testing if ${a}^{2}+{b}^{2}$ is the same as ${c}^{2}$.

So, the theorem is used to find the sides of a right triangle, and to check whether a given triangle is a right triangle.

Example 1

You have a right triangle where $AB=3$, $BC=5$ and $\angle A=\text{}90\text{}\text{°}$. What is the length of $AC$?

The hypotenuse is always the side of a right triangle opposite to the right angle—which is $90$°. You know, therefore, that $AC$ is a leg of the triangle. You put this into the formula and find that $\begin{array}{llll}\hfill A{B}^{2}+A{C}^{2}& =B{C}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {3}^{2}+A{C}^{2}& ={5}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 9+A{C}^{2}& =25\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill A{C}^{2}& =25-9\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill A{C}^{2}& =16\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill AC& =\sqrt{16}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill AC& =4\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

The length of side $AC$ is 4.

Example 2

You want figure out if a triangle with sides 5, 8 and 9 is a right triangle.

You know that the hypotenuse is the longest side of a right triangle, so you find the square of the longest side:

 ${9}^{2}=81$

Then, you find the sum of the squares of the other sides of the triangle

 ${8}^{2}+{5}^{2}=64+25=89$

Since $81\ne 89$, you know for sure that this isn’t a right triangle.