Pythagoras' theorem

How high does a ladder reach when it leans against a wall? How much shorter is the diagonal path across the park compared to walking around it? Questions like these are answered by a theorem that is about 2500 years old and still lives inside every navigation app, every construction plan and every video game: Pythagoras' theorem.

Spotting right triangles

The theorem only works for right triangles, that is, triangles with a 90° angle. The two sides that form the right angle are called the legs. The longest side lies opposite the right angle and is called the hypotenuse. Pythagoras says: the squares on the two legs together are exactly as large as the square on the hypotenuse. As a formula: a² + b² = c², where c is the hypotenuse.

Finding the missing side

If you know both legs, you find the hypotenuse in two steps: square and add first, then take the square root. The root undoes the squaring, √25 = 5 simply means that 5 · 5 = 25. Example with a = 3 and b = 4: we get 3² + 4² = 9 + 16 = 25, so c = √25 = 5. If a leg is missing instead, rearrange the formula: a² = c² − b². With c = 10 and b = 8 that gives a² = 100 − 64 = 36, so a = √36 = 6. Side lengths are never negative, so only the positive root counts here.

Pythagoras in everyday life

Wherever there is a right angle, Pythagoras is close by. A ladder against a wall forms a right triangle with the ground and the wall, and the ladder itself is the hypotenuse. The path straight across a rectangular park is the hypotenuse of a triangle whose legs are the length and the width of the park. And if you want to check whether a corner is truly square, measure a stretch along each of the two walls plus the distance between the two end points. Of these three lengths the largest goes into the formula as c: if a² + b² = c² holds, the angle is exactly 90°.