The Pythagorean Theorem — Rearranging Pieces to a²+b²=c²
Fill one big square two ways with four identical right triangles: once leaving a² and b², once leaving c². Equal leftovers ⟹ a²+b²=c².
Problem
Prove a² + b² = c² for a right triangle by rearranging the same four triangles inside one square.
Cut · rotate · rearrange
Press Play or move Next one step at a time. Follow the pieces until the answer becomes visible. (Arrow keys and Space also work.)
A square of side a+b. Four equal right triangles leave gaps forming two squares, a² and b².
Why this approach works
Don’t compute areas — compare two arrangements of the same pieces. In a square of side a+b, place four right triangles so the gaps form two squares a² and b². Translate those four triangles into the four corners, and the gap becomes one square c² on the hypotenuse. Same big square, same four triangles, so the leftovers are equal: a² + b² = c². Algebraically, (a+b)² − 4·½ab = c².
Proof
Comparing two ways to fill an (a+b)-square with the same four right triangles: one leaves a²+b², the other leaves c². The big square and triangles are identical, so the leftovers are equal — a²+b²=c².
Try it yourself
Subtract the four triangles (each ½ab) from (a+b)²: the 2ab cancels, leaving a²+b² — which equals c².
Related mathematics
Beyond MathVoyage
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