How to use the pythagorean Theorem Calculator
- Choose which side you want to find: the hypotenuse c, or one of the legs a or b.
- Enter the two sides you know. The hypotenuse is always the longest side, opposite the right angle.
- Use any unit you like, such as cm, m, inches or feet; the answer is in the same unit.
- The result updates instantly as you type. There is no button to press.
- Use Copy result to copy the figures, or Copy link to share a link that reopens the pythagorean Theorem Calculator with the same inputs.
How it works
The Pythagorean theorem says that in any right triangle, the square of the hypotenuse (the side opposite the 90° angle) equals the sum of the squares of the other two sides: a² + b² = c². It is named after the Greek mathematician Pythagoras, though the relationship was known in Babylon and India centuries earlier, and it has hundreds of known proofs.
It is one of the most practical results in maths. Builders use the 3-4-5 rule to check that corners are square. It gives the diagonal size of a TV screen or a room, the length of a ladder needed to reach a window, the straight-line distance between two points on a map or a grid, and the length of a ramp or roof rafter.
Given any two sides, this calculator finds the third, then uses trigonometry to give the two acute angles, which always add up to 90°. It also gives the area (half the product of the legs), the perimeter and the height from the right angle to the hypotenuse. When you are finding a leg, the hypotenuse must be longer than the other leg, or no right triangle exists.
Formula
Area = a × b ÷ 2. Height to the hypotenuse = a × b ÷ c.
Example
A right triangle has legs of 3 and 4. The hypotenuse is √(9 + 16) = √25 = 5. Its angles are about 36.87° and 53.13°, the area is 6 and the perimeter is 12. In practice, a 10-foot ladder whose foot stands 6 feet from a wall reaches √(100 − 36) = 8 feet up the wall.
Frequently asked questions
Does the Pythagorean theorem work for all triangles?
No, only right triangles. For other triangles, use the law of cosines: c² = a² + b² − 2ab·cos(C), which reduces to Pythagoras when C = 90°.
What is a Pythagorean triple?
A set of three whole numbers that satisfy a² + b² = c², such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any multiple of a triple, such as 6-8-10, is also a triple.
How do I check if a corner is square?
Measure 3 units along one side and 4 along the other. If the diagonal between those marks is exactly 5, the corner is 90°. Use 6-8-10 or 9-12-15 for bigger spaces.