The Pythagorean theorem

​​ ​This is Pythagoras another one of the ancient Greeks. He lived a couple of centuries before Euchlid, around 2500 years ago on the island of Samos in the Greek archipelago in the Agean sea. Pythagoras was a mathematician and philosopher. Today he's best known for the Pythagorean theorem. This is a mathematical formula that describes how the lengths of the three sides of the right triangle relate to each other.

Evidence exists that this relationship was already known in Greece when Pythagoras wrote it down and mathematicians in China and in India probably discovered it on their own, perhaps even earlier, but we still call this formula the Pythagorean theorem. ​ ​This is what it looks like if you draw it, the Pythagorean theorem, start with a right triangle. Draw a square on each side If you add the areas of the two smaller squares together, you get the same area as the largest square has. Pause the video and count the boxes to see for yourself. The largest square is 5x5, 25 boxes. The middle one is 4x4,16 boxes and the smallest one is 3x3, 9 boxes and 25 is equal to 16 + 9.

Or, more generally, the square of the longest side equals the sum of the squares of the two shorter sides. This works for all right triangles. Here are two definitions you should know about. The longest side of a right triangle is called the hypotenuse and the two shorter sides are called legs. We often denote the hypotenuse by the letter C and the two legs by the letters A and B.

We can now write the Pythagorean Theorem as C squared equals A squared plus B squared. ​ ​To use the Pythagorean Theorem, you need to be able to solve equations. Here is a right triangle where we know the lengths of both legs but not the hypotenuse. Put the values into the Pythagorean Theorem. C squared equals 5 squared plus 12 squared. Calculate the squares and add them.

Now comes the interesting part. Where do you move from here? The hypotenuse squared is 169. How do you solve an exponential expression? The opposite of exponential expressions are root expressions, so we take the square root of both sides of the equation.

The square root of C squared equals the square root of 169. The square root of C squared is C. This is why we used the root expression . Find the square root of 169 on your calculator, and you get 13. The hypotenuse C is 13. ​ ​The Pythagorean Theorem works on all right triangles, but not on any other types of triangles.

The square of the longest side, the hypotenuse, equals the sum of the square of the two shorter sides, the legs. To solve a problem with the Pythagorean Theorem plug in the values you have and solve the equation. Take the square root from both sides after you have isolated the unknown values squared on one side. Here's a bonus, take a right triangle and give both legs the length one. Then the hypotenuse is an irrational number, the square root of two.

This is what Hypaspists discovered and what made the Pythagoreans extremely annoyed because they believed that all numbers had to be rational. Watch the lesson on 'Irrational Numbers' to find out more.