The Pythagorean Theorem Explained (with Examples)
The Pythagorean theorem is one of the most useful rules in maths. It connects the three sides of a right-angled triangle, and once you get it, a whole range of problems become easy.
The theorem
In any right-angled triangle, the square of the longest side (the hypotenuse, opposite the right angle) equals the sum of the squares of the other two sides.
As a formula: a² + b² = c², where c is the hypotenuse.
Worked example
A right triangle has legs of 3 and 4. Find the hypotenuse.
a² + b² = c² → 3² + 4² = c² → 9 + 16 = 25 → c = √25 = 5. The hypotenuse is 5.
Finding a shorter side
If you know the hypotenuse and one leg, rearrange. Say c = 13 and a = 5: c² − a² = b² → 169 − 25 = 144 → b = √144 = 12.
Where you will use it
Distances on a map or graph, the diagonal of a screen or room, construction, navigation — anywhere a right angle appears, the theorem can find a missing length.
Frequently asked questions
Does the Pythagorean theorem work for all triangles?
No — only right-angled triangles. For other triangles you need the law of cosines.
What is the hypotenuse?
The longest side of a right triangle, always opposite the right angle.