What this calculator does
Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It is one of the oldest and most-used results in geometry, turning up anywhere a right angle and a straight-line distance meet.
This calculator solves the theorem in the direction you actually need: either the hypotenuse from two known legs, or a missing leg from the hypotenuse and the other leg. Switching the "solve for" option changes which formula runs, but the working is shown either way.
The formula
To find the hypotenuse from two legs, c = √(a² + b²). To find a missing leg when the hypotenuse and one leg are already known, rearrange the same theorem to leg = √(c² − b²), which only works when the hypotenuse is longer than the known leg, as it always must be in a real right-angled triangle.
| Term | Meaning |
|---|---|
| a, b | The two legs of the right-angled triangle, meeting at the right angle. |
| c | The hypotenuse: the longest side, opposite the right angle. |
| a² + b² = c² | Pythagoras theorem: the sum of the squares of the two legs equals the square of the hypotenuse. |
The inputs explained
| Field | What to enter |
|---|---|
| Solve for | Choose whether you are solving for the hypotenuse (from two legs) or for a missing leg (from the hypotenuse and one leg). |
| Known side a (leg, or hypotenuse if solving for a leg) | A leg length, or the hypotenuse if solving for a missing leg. |
| Known side b (a leg) | The other known leg. |
When to use it
Squaring up a frame or foundation
Builders use Pythagoras theorem to check a corner is a true right angle, by confirming the diagonal (hypotenuse) matches √(a² + b²) for the two measured sides.
Finding a diagonal distance
The straight-line distance across a rectangular block, screen or plot of land, when only the two straight sides are known, is exactly the hypotenuse of a right-angled triangle formed by those sides.
Working out a ladder or ramp length problem
Given a height to reach and a horizontal base distance, the hypotenuse gives the minimum ladder or ramp length needed; given a fixed ramp length and one dimension, solving for the missing leg gives the other.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Pythagoras theorem formula: solving for the hypotenuse
A fixed leg of 4, across a range of second-leg lengths, solving for the hypotenuse.
Solving Pythagoras theorem for a missing leg
A fixed known leg of 3, across a range of hypotenuse lengths, solving for the missing leg.
Questions
What is Pythagoras theorem?
Pythagoras theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
What is the Pythagoras theorem formula for the hypotenuse?
c = √(a² + b²), where a and b are the two legs meeting at the right angle and c is the hypotenuse, the side opposite that right angle.
Can Pythagoras theorem find a missing leg instead of the hypotenuse?
Yes. Rearranging a² + b² = c² gives leg = √(c² − b²), used when the hypotenuse and one leg are known and the other leg is missing.
Does Pythagoras theorem work for any triangle?
No, only for right-angled triangles. For a triangle without a right angle, the law of cosines is the general version that reduces to Pythagoras theorem when the angle is exactly 90°.
For every measurement of a right-angled triangle, including area and both acute angles, from two known legs, see the right triangle / Pythagoras calculator. For a triangle with no right angle, see the triangle from three sides calculator.