Why does a² + b² = c² work for every right triangle?
Squares drawn on a right triangle's two short sides always add up to the square on its long side. People have proved it in at least 370 ways.
▶ Start the storyTake any triangle with a right angle in one corner. Draw a square on each of its three sides. The Pythagorean theorem says that the two smaller squares, put together, have exactly the same area as the big square on the longest side, the hypotenuse. In letters: a² + b² = c². A 3-4-5 triangle shows it: 9 + 16 = 25.
Why must that be true for every right triangle? One way to see it is a rearrangement. Take a big square of side a + b and fit four copies of your triangle inside it in two different ways. In one arrangement they leave a tilted square of side c in the middle. In the other they leave two squares, one of side a and one of side b. The big square and the four triangles are identical in both pictures, so what is left over must match: c² on one side, a² + b² on the other.
Step 1: Draw two squares of side a + b
Same size, same area
Step 2: Add four copies of the triangle to each
Each triangle has legs a and b, hypotenuse c
Step 3: Arrangement one leaves a tilted square
Its side is c, so its area is c²
Step 4: Arrangement two leaves two squares
One of area a², one of area b²
Step 5: Same leftover space
So c² = a² + b²
People never tired of finding new routes to the same fact. The theorem has been proved by many different methods, possibly more than any other, and one book collects 370 of them. Some use similar triangles, whose sides keep the same ratios at any size. One was published by James A. Garfield while he was a congressman, before he became president of the United States.
The name is a bit of a misdirection. Historians conclude that the rule was in widespread use in Babylonian times, over a thousand years before Pythagoras was born, and the Greek writings from the five centuries after him do not credit him with any specific discovery. The oldest surviving proof that builds it carefully from axioms is Euclid's, around 300 BC.
The distance formula in coordinates is derived from the theorem. And it carries a secret: it holds only on a flat surface.
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Recap
Two arrangements of the same four triangles inside a square of side a + b leave c² in one picture and a² + b² in the other.
💡 A trick to remember it · Same square, same four triangles, different leftovers: one gap of c², or two gaps of a² and b².
Surprising fact · The rule was in widespread use in Babylon over a thousand years before Pythagoras was born.
Sources (1)
No source, no claim. Every fact in this lesson (15 claims) cites at least one of these.