The pythagorean theorem is a formula relating the sides of a right triangle derived by a Greek mathematician named Pythagorus in the 500s BC. The formula works for all right triangles with sides of a, b, and c where c is the hypotenuse, or side opposite the right angle.

The formula he derived was known as the pythagorean theorem:

a^2+b^2=c^2 (The ^2 means an exponent of 2, or squared).

This is read as a squared + b squared = c squared.

For example, if there was a right triangle with the sides of 3, 4, and 5, then a would be 3, b would be 4, and c would be 5.

Plug this into the formula, and it works!

3^2+4^2=5^2. Simplifying the squares, we get:

9+16=25. Simplifying by addition, we get the true statement:

25=25.

The pythagorean theorem can be applied in numerous ways, and here I'll give an example. Let's say you have a TV and you need to find the length of the diagonal going across. You know that the sides are exactly 3 feet by 5 feet. It doesn't matter which you make a or b, so let's make:

a=3, b=5. We want to find c, the unknown variable, the diagonal. Plug this into the equation and get:

3^2+5^2=c^2. Simplify the squares and get:

9+25=c^2. Simplify this using addition and get:

34=c^2. Now, we need to solve for c. C squared equals 34, so we need to take the square root of 34. Using a calculator, we can approximate:

c=5.83095...

So, using the Pythagorean theorem, we found that the hypotenuse was about 5.8 feet!

Hope this helped, thanks!

Here's a good visualization of the Pythagorean Theorem: