The Pythagorean Theorem
a² + b² = c², geometry's most ancient equation
Summary
In a right triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse. Adjust sides a and b with the sliders and watch the areas of the three squares always satisfy a² + b² = c².
In depth
Although it is named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), this theorem was known to the Babylonians more than a thousand years before him. Its modern statement is simple: in a right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides. The visual proof: draw a square on each side of the triangle, and the areas of the two smaller squares add up to the area of the large one. It is not just a formula but the defining character of Euclidean geometry: it is the rule for measuring distance in space. It also lives on in modern physics: a version of it works in flat spacetime, but curved spacetime (general relativity) requires modifications.
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