The Pythagorean Theorem

The theorem is really a statement about area. Build a square on each side of a right triangle. The two squares sitting on the legs add up, in area, to exactly the square sitting on the hypotenuse. You can see why with a classic rearrangement: take a big square with side (a+b), and place four copies of the right triangle inside it, pinwheel style. The leftover space in the middle is a square of side c, with area c^2. Rearrange the same four triangles inside the same big square a different way, and the leftover space splits into two squares, with areas a² and b^2. Same big square, same four triangles, so the leftover areas must be equal: a² + b² = c^2.

Before plugging in numbers, find the hypotenuse. It is always the longest side, and it always sits opposite the right angle, never adjacent to it. Mislabeling the hypotenuse as a leg is the single fastest way to get a right-triangle problem wrong, so look for the right angle mark first, then name the side across from it c.

This theorem only works on right triangles, because the proof depends entirely on that 90 degree angle to make the pinwheel rearrangement close up into neat squares. A triangle with angles 80, 60, and 40 has no right angle, so there is no hypotenuse and no a² + b² = c² relationship; you would need the law of sines or law of cosines instead. The theorem is the gateway to the rest of right-triangle trigonometry: the two special right triangles (45-45-90 and 30-60-90) are just the Pythagorean theorem with extra angle information, and sine, cosine, and tangent all describe ratios between sides that still obey a² + b² = c^2.