There's a single equation, written down twenty-five centuries ago,
that still shapes how we measure the world.
Start with any right triangle. Three sides, three corners, and
one angle that's exactly ninety degrees — the small square
corner that makes everything else fall into place.
Call the two shorter sides a and b. The longest side, lying opposite the
right angle, has its own name. We call it the hypotenuse — c.
Pythagoras's claim is striking in its simplicity. The squares
built on the two short sides, added together, equal
the square built on the long one.
Build a literal square on each side. One with area a-squared.
One with area b-squared. One with area c-squared. The claim is
that the two small squares, combined, fill the largest one exactly.
It sounds too clean to be true. So let's prove it without
any algebra — just by moving shapes around inside two boxes.
Take two identical squares, each with side length a plus b.
They cover the same area — that part isn't up for
debate. We're going to fill them in two different ways.
In the first square, drop four copies of our triangle into the
corners. What's left over is two perfect squares — one of side
a, one of side b. Total leftover: a-squared plus b-squared.
In the second, place the same four triangles — this
time rotated, so their hypotenuses meet at the center. What
remains is a single tilted square, with side exactly c.
Both big squares started with the same area. Both swallowed
up the same four triangles. So whatever's left over in
each one must also have the same area.
Which gives us the equation. The two leftover squares on one side. The
one tilted square on the other. Equal in area, by pure geometry.
a-squared plus b-squared equals c-squared. Not a numerical coincidence
— a geometric necessity, written into the shape itself.
Try it with any right triangle you like. Any side
lengths, any orientation. The relationship holds, every single time.
This is the Pythagorean theorem. Two and a half thousand years old,
and still the cleanest, most stubborn fact in all of geometry.