3² + 4² comes out to exactly 5² — no more, no less. That's not a coincidence, and you don't have to take it on faith: four triangles sliding around inside one square frame will show you, with your own eyes, why it has to be true.
The props never change: one big frame, four identical triangles.
Legs a=3 and b=4, hypotenuse c. Each triangle has area ½ab = 6, so the four together cover 24. Add one big square of side a+b=7, area 49. That's every prop we'll use.
The four right angles tuck into the frame's four corners, and the four hypotenuses fence off a hole in the middle. Every side of the hole is a hypotenuse, length c; wherever the hole touches the frame, the two acute angles of neighboring triangles meet — and since they're the complementary pair from a single right triangle, they add up to exactly 90°, so the angle left for the hole is exactly 90° too. The hole is a square of side c, area c². Check the numbers: 49 − 24 = 25.
Keep the frame where it is and slide three of the triangles over (no rotation needed). They pair up into a×b rectangles tucked into two opposite corners, and the leftover space becomes two squares, b×b and a×a: 16 + 9 = 25.
Same big frame, minus the same four triangles — however you arrange them, what's left over is 49 − 24. So c² = a² + b². And nothing depended on "3" and "4": take any right triangle, swap in its a and b, and the same two arrangements work just as well.
That's the Pythagorean theorem. Whole-number combinations that satisfy it are called Pythagorean triples: 3-4-5, 5-12-13, 8-15-17… The 3-4-5 triangle is the smallest — the ancient Chinese text Zhoubi Suanjing recorded exactly this triple ("gou 3, gu 4, xian 5") centuries before Euclid.
Tie 12 evenly spaced knots in a rope, pull it into a triangle with sides 3, 4, and 5, and the corner facing the 5 is a perfect right angle. Legend has it Egyptian rope-stretchers used exactly this trick to lay out the pyramids. Now you know why it works: 3² + 4² = 5².