The Pythagorean Theorem says that a^{2}+b^{2}=c^{2}.

In math classes, the problems you’d have to solve using the Pythagorean Theorem sound something like…

A painter has a 17-foot ladder and in order to reach the spot that he needs to paint, the top of his ladder needs to rest 15-feet up the wall. How far must the bottom of his ladder be from the base of the wall for **BLAH BLAH BLAH**?

**Every time I see these kinds of problems, I think they’re absolutely ridiculous!** If I’m a painter, I’m not going to measure the distance my ladder needs to be from the wall, I’m just gonna rest it against the wall and move it around until it’s in the right position! AND nobody uses ladders that rest against a wall anymore, we use the ladders that stand on their own. If you’re not a physicist or a carpenter, you won’t need to use the Pythagorean Theorem, so let’s stop making word problems that lie to you.

I bet if I asked someone what the Pythagorean Theorem is, they would say: “if I know the lengths of two sides of a right triangle, I can use the Pythagorean Theorem to find the third length.”

But try thinking about it like this:

Imagine wanting so desperately to understand the world you live in, but you have no formulas to work with. You have no idea how to think about the space things occupy in two dimensions let alone in three. Some philosopher walks up to you and says that if you have two squares that take up different amounts of space, then there is always a square that you can create that takes up the same amount of space as those two squares combined. And if you line these squares up so that their corners are touching, they will always make a triangle. And not just any triangle, no matter what size your three squares are, one corner will be the same size.

It sounds a little more complicated now, doesn’t it? If you don’t understand what he’s saying, or if it doesn’t seem obviously true, wouldn’t you want him to prove it? You wouldn’t just take his word for it, would you? So… why do we all just accept it in our math classes?

The Pythagorean Theorem is simple mathematically, but it says so much about area, space and what it means to square a number.

Below is a video of my favorite proof of the Pythagorean Theorem (it’s only 45 seconds and you don’t need sound). This proof is so beautiful and simple that I don’t even need to explain what’s happening for you to * see* how it proves the Pythagorean Theorem (or at least one case in which it works).

But what I love most about this proof (because I’m a nerd) is that in order to use water inside of boxes to prove the Pythagorean Theorem, you’ve started talking about three dimensions, instead of two. Because water occupies three-dimensional space. So really this proof says that h(a^{2})+h(b^{2})=h(c^{2}). This slight difference, which doesn’t change the theorem mathematically, says WORLDS about how we can think about shapes and space.

The next post will get even deeper into this topic to talk more about how to think about space, area and what it means to square something!