The Pythagorean theorem — Mathematics, 14–17 years
A way to connect the three side lengths of a right-angled triangle. Mathematics, 14–17 years.
The longest side
In a right-angled triangle, the two shorter sides meet at the right angle. The longest side lies opposite it and is called the hypotenuse. Their lengths obey a² + b² = c²: the squares on the two shorter sides have the same total area as the square on the hypotenuse.
The problem it solves
Measuring a diagonal directly can be awkward, while two perpendicular lengths may be easy to know. This theorem finds the missing distance from those two lengths. It was known in ancient Babylon and India and later linked to Pythagoras, so it is a practical relationship, not a rule chosen at random.
A ladder against a wall
A ladder reaches 12 m up a wall, with its foot 5 m from the wall. The wall, ground and ladder form a right triangle. Let c be the ladder: c² = 5² + 12² = 25 + 144 = 169. Since √169 = 13, the ladder is 13 m long.
Which side is c?
The usual mistake is putting the longest side in the wrong place or calling any side c. That is understandable because diagrams can be rotated and letters are arbitrary. In a right triangle, c must be the side opposite the right angle; it is always the longest side.
Distances on a map
Builders use this relationship to check whether a corner is truly square: measured lengths of 3 m, 4 m and 5 m satisfy 3² + 4² = 5². It also estimates a straight-line distance when east–west and north–south changes are known. It applies only when those directions are perpendicular.
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