The Pythagorean Theorem and the Square Root of 2

Most of you probably remember the Pythagorean Theorem, and the rest of you probably remember that there was a time when you remembered it. The Pythagorean Theorem relates the lengths of the sides of a right triangle. If and are the lengths of the shorter sides (the legs) of a right triangle and is the length of the longest side (the hypotenuse), then .

In other words, if you draw three squares based on the side lengths of the triangle, the total area of the smaller two is equal to the area of the largest.

The Pythagorean Theorem is named for Pythagoras, a Greek mathematician. To be fair, he was not the only person to do so. This fact, so fundamental to geometry and measurement, was known independently to many ancient cultures, and in some places there is evidence that the fact was known long before Pythagoras.  I like to think that the theorem bears Pythagoras’ name because he is the most colorful choice for a namesake — mathematician, numerologist, leader of a secret society (yes, really!).

What you might not expect is that it is actually quite simple to see why the Pythagorean Theorem is true. You don’t have to be an ancient mathematician shrouded in mystery to discover it.  Actually all it takes is some paper squares cut into colored pieces.

(How this illustrates the Pythagorean Theorem is explained after the jump . . . but try to see it for yourself before reading on.)