What the theorem states
In any right triangle — one with a 90° corner — the side opposite the right angle is the hypotenuse, and it is always the longest side. The other two sides are called the legs.
The Pythagorean theorem says the square built on the hypotenuse has exactly the same area as the two squares built on the legs combined: a² + b² = c². This single relationship lets you find any one side once you know the other two.
Solving for the side you leave blank
The calculator looks at which two boxes you filled and rearranges the formula automatically:
- Both legs given: the hypotenuse is the square root of (a² + b²).
- A leg and the hypotenuse given: the missing leg is the square root of (c² − a²).
- It rejects a hypotenuse that is not longer than each leg, because that triangle cannot exist.
Where you will use it
The theorem appears far beyond textbooks. Builders use the 3-4-5 rule to check that a corner is square, navigators turn east-west and north-south travel into a direct distance, and the same idea underlies the coordinate distance formula.
A practical tip: the theorem only holds for right triangles. If your triangle has no 90° angle, use the law of cosines (the general triangle solver) instead.
Formula
c = √(a² + b²)