The Pythagorean Theorem Explained

The Theorem

In a right angled triangle the square of the longest side equals the sum of the squares of the other two sides. Written as a squared plus b squared equals c squared, where c is the hypotenuse.

Finding the Hypotenuse

Add the squares of the two legs and take the square root. Legs of 5 and 12 give the square root of 25 plus 144, which is the square root of 169, which is exactly 13.

Finding a Missing Leg

Subtract the square of the known leg from the square of the hypotenuse, then take the square root. With a hypotenuse of 13 and one leg of 5, the other leg is the square root of 169 minus 25, which is the square root of 144, which is 12.

Integer Triples

Some whole number triples satisfy the theorem exactly: 3, 4, 5 and 6, 8, 10 which is the same triple scaled, and 5, 12, 13, 8, 15, 17 and 9, 12, 15. There are infinitely many, generated by a formula due to Euclid.

Where It Applies Beyond Triangles

Coordinate geometry uses it to find distance between points, and three dimensions extend it by adding a third squared term. Surveying, navigation and computer graphics all rely on the same idea.

Where It Does Not Apply

It is specific to right angled triangles. Applying it to an acute or obtuse triangle gives the wrong answer, and there is no simple equivalent for general shapes, which is why general geometry needs a different tool.