Proof of the Pythagorean Theorem

This month, I decided I would focus on one of the most famous theorems in all of mathematics: The Pythagorean Theorem. Though it sounds kind of dull on the face, it has a lot of extremely interesting things within it.

First, let me go over what it is. If you take a right triangle, and label the measure of its shortest side a, its middle side b, and its longest side c, then you can count on the fact that:

a² + b² = c² 

This can be used in geometry problems, like to find the longest side, or the hypotenuse, of a triangle with shorter sides, or legs, 3 and 4. You would simply plug 3 in for a, 4 in for b, and solve for c.

(3)² + (4)² = c² 

9 + 16 = c² 

25 = c² 

5 = c

The first thing I wondered when seeing this theorem is why it is true. It seems odd that any right triangle’s measurements must fit this criteria. And I never found a proof I liked until extremely recently. So, I would like to share it.

Take the following diagram:

We are going to try to figure out the area of this square. The simplest way to do it would probably be to square the square’s side. This length would be a + b.

(a + b)²

a² + 2ab + b²

Another way we could get this result is to find the area of the inside square, and then the area of the four triangles surrounding it. The inside square has length c, and the outer triangles have a base of b and height of a. So, we would get:

c² + 4(ab/2)

c² + 2ab

Since these two quantities are both the area of the same square, we can set them equal to each other. After simplification, we would get:

a² + 2ab + b² = c² + 2ab

a² + b² = c²

And we end up with our Pythagorean Theorem. I was so intrigued to see that this proof was so quick and simple, as well as pretty interesting.