Math & Money

How many ways are there to prove the Pythagorean theorem?

What do Euclid, twelve-year-old Einstein, and American President James Garfield have in common? They all came up with elegant proofs for the famous Pythagorean theorem, the rule that says for a right triangle, the square of one side plus the square of the other side is equal to the square of the hypotenuse. In other words, a²+b²=c². This statement is one of the most fundamental rules of geometry, and the basis for practical applications, like constructing stable buildings and triangulating GPS coordinates.

The theorem is named for Pythagoras, a Greek philosopher and mathematician in the 6th century B.C., but it was known more than a thousand years earlier. A Babylonian tablet from around 1800 B.C. lists 15 sets of numbers that satisfy the theorem. Some historians speculate that Ancient Egyptian surveyors used one such set of numbers, 3, 4, 5, to make square corners.

The theory is that surveyors could stretch a knotted rope with twelve equal segments to form a triangle with sides of length 3, 4 and 5. According to the converse of the Pythagorean theorem, that has to make a right triangle, and, therefore, a square corner. And the earliest known Indian mathematical texts written between 800 and 600 B.C. state that a rope stretched across the diagonal of a square produces a square twice as large as the original one. That relationship can be derived from the Pythagorean theorem.

But how do we know that the theorem is true for every right triangle on a flat surface, not just the ones these mathematicians and surveyors knew about? Because we can prove it. Proofs use existing mathematical rules and logic to demonstrate that a theorem must hold true all the time. One classic proof often attributed to Pythagoras himself uses a strategy called proof by rearrangement. Take four identical right triangles with side lengths a and b and hypotenuse length c.

Arrange them so that their hypotenuses form a tilted square. The area of that square is c². Now rearrange the triangles into two rectangles, leaving smaller squares on either side. The areas of those squares are a² and b². Here’s the key. The total area of the figure didn’t change, and the areas of the triangles didn’t change. So the empty space in one, c² must be equal to the empty space in the other, a² + b². Another proof comes from a fellow Greek mathematician Euclid and was also stumbled upon almost 2,000 years later by twelve-year-old Einstein. This proof divides one right triangle into two others and uses the principle that if the corresponding angles of two triangles are the same, the ratio of their sides is the same, too.

So for these three similar triangles, you can write these expressions for their sides. Next, rearrange the terms. And finally, add the two equations together and simplify to get ab²+ac²=bc², or a²+b²=c². Here’s one that uses tessellation, a repeating geometric pattern for a more visual proof. Can you see how it works? Pause the video if you’d like some time to think about it. Here’s the answer.

The dark gray square is a² and the light gray one is b². The one outlined in blue is c². Each blue outlined square contains the pieces of exactly one dark and one light gray square, proving the Pythagorean theorem again. And if you’d really like to convince yourself, you could build a turntable with three square boxes of equal depth connected to each other around a right triangle. If you fill the largest square with water and spin the turntable, the water from the large square will perfectly fill the two smaller ones. The Pythagorean theorem has more than 350 proofs, and counting, ranging from brilliant to obscure. Can you add your own to the mix?

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