How to Prove the Pythagorean Theorem

What does the Pythagorean Theorem tell us about triangles? Is it always true?  And how can you prove this famous theorem arts-and-crafts style? Keep on reading to find out.

Jason Marshall, PhD,
June 10, 2016
Episode #282

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Pythagorean Theorem on Chalkboard

It’s summertime here in the northern hemisphere, which means it’s that special time of year when parents of school-aged children are frantically looking for stuff for their kids to do.

If that describes your life right now, I have good news for you. Because today we’re talking about an arts-and-crafts inspired project that you and your kids can do together. It’s sure to keep you busy for at least a little while and it’ll even teach everybody a thing or two about mathematical thinking along the way. And, of course, it’ll be all kinds of good times.

So, without further ado, let’s get started.

What Is the Pythagorean Theorem?

As we’ve talked about previously, the famous Pythagorean Theorem can be expressed in equation form as a2 + b2 = c2—where a, b, and c are the lengths of the sides of a right triangle. As we’ve also talked about, the point of this (or any other) equation is to say that the combination of variables in the expression on the left side of the equals sign must have the same value as the combination of variables in the expression on the right.

At least that’s the algebraic way of thinking about things. And while this view is all well and good—and indeed a very useful way of viewing the world—it’s also good to keep in mind that it’s not the only possible view. For example, we can also picture the meaning of the Pythagorean Theorem graphically.

How to Think About the Pythagorean Theorem

To see how a picture can help us understand the Pythagorean Theorem, let’s draw one. In particular, let’s draw a classic 3–4–5 right triangle with legs that are 3 and 4 units long (so a = 3 and b = 4) and a hypotenuse whose length is given by the square root of a2 + b2. So c = √25 = 5. I encourage you to engage your crafty side from here on out, to bust out a pencil and paper, and to start sketching things out. Here’s my version of the drawing to get you started.

Pythagorean Triple


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