Visual Proof of the Pythagorean Theorem
Why does a² + b² = c²?
The Pythagorean Theorem, undoubtedly one of the most famous formulas there is. A classic among high school classrooms. A pillar of mathematics education. Surely, you’re familiar with it?
But have you stopped to think about why it works? Why does a² + b² = c²?Or have you just accepted it?
If you’re anything like me, algebraic arguments are alright, but they’re not really a home run. I mean if I want something to really sink in, and I mean really, really sink in I need to see why it’s true and really understand the mechanics.
So for all you visual learners out there, you’re in luck because the visual proof of the Pythagorean Theorem is as insightful as it is elegant.
Visual Proof 👊🏼
Step 1
Start by making one big square, then place two smaller squares inside it.
It doesn’t really matter what size these smaller squares are, all that matters is that when side by side they’re the same length as one side of the big square. Something like this for instance:
I’m going to call the side lengths of the maroon square a and the side lengths of the tan square b. That means the maroon square has an area of a² and the tan square has an area of b².
Step 2
Let’s section out the remaining area of the big square into rectangles.
We know that the side length of our large square is a+b. This is handy because it can help us figure out the missing lengths of the rectangles.
Once we fill in those missing lengths, we realize we have two identical rectangles with areas of ab.
So the area of the large square, (a+b)², is equal to sum of the areas of the four shapes inside it: a² + b² + ab + ab, or equivalently:
Now let’s do some rearranging.
Step 3
First draw a diagonal in each rectangle. Since the two rectangles are identical, their diagonals will be the same length. Let’s call that length c.
Now here’s the tricky part. I’m going to remove the maroon and tan squares and rotate the two inner triangles so that they’re arranged along the edge of the big square as well.
That leaves me with this new arrangement:
In the process of rotating the two inner triangles we have made a new square with side lengths c and an area of c².
We have also created a new way of describing the area of (a+b)². It is now the sum of: c² + 4 triangles of area 1/2(ab).
Which can be simplified to:
Step 4
We now have two formulas for (a+b)², which means we can set equation 1, from earlier, and equation 2 equal to one another.
Lastly, subtract 2ab from both sides, and look there’s the Pythagorean theorem!
Looking back at our last square diagram, we see that the right triangles have legs a and b, with a hypotenuse of c as anticipated.
Mystery solved! Now we know why the sum of the square of the legs of a right triangle equal the square of the hypotenuse. Pretty clever stuff :)
Thanks for reading!
