Pythagorean Theorem Calculator
Solve for a missing side of a right triangle — the hypotenuse or either leg — using the Pythagorean theorem, given the other two known sides.
Calculator Inputs
Choose which side of the right triangle you want to solve for.
Meaning depends on the mode above: leg a for "Solve for hypotenuse (c)", or the hypotenuse (c) for "Solve for leg a" and "Solve for leg b".
Meaning depends on the mode above: leg b for "Solve for hypotenuse (c)" and "Solve for leg a", or leg a for "Solve for leg b".
- Missing Side
- 5.00
- Formula Used
- c = √(a² + b²)
About these results
CalcSpectrum calculators produce automated estimates using established formulas and the specific values you enter. Because real-world outcomes depend on input accuracy, mathematical assumptions, and rounding conventions, please verify critical figures with an appropriately qualified financial, tax, medical, legal, or other professional, as relevant.
How It's Calculated
Formula
The Pythagorean theorem describes a fixed relationship between the three sides of any right triangle (a triangle with one 90-degree angle): the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides, called legs. Written as a formula, a² + b² = c², where a and b are the legs and c is the hypotenuse. Because this is an equation with three variables, knowing any two of the three sides lets you solve for the third algebraically. Solving for the hypotenuse (c = √(a² + b²)) is always possible for any two positive leg lengths — there is no additional constraint between two legs. Solving for a leg (a = √(c² − b²), or b = √(c² − a²)) requires the known hypotenuse to be strictly greater than the known leg, because the hypotenuse is always the longest side of a right triangle; if the "hypotenuse" you enter is not actually longer than the known leg, the two values cannot describe a real right triangle, and this calculator reports that clearly instead of computing a nonsensical result.
Worked Examples
Solve for the hypotenuse: legs 3 and 4 (the classic 3-4-5 triangle)
- Formula: c = √(a² + b²)
- Substitute: c = √(3² + 4²) = √(9 + 16)
- c = √25 = 5
- The hypotenuse is 5.
Solve for a leg: hypotenuse 5 and leg b 4
- Formula: a = √(c² − b²)
- Check the hypotenuse is longer than the known leg: 5 > 4, so this is a valid right triangle.
- Substitute: a = √(5² − 4²) = √(25 − 16)
- a = √9 = 3
- The missing leg is 3.
Frequently Asked Questions
How is this different from the Triangle Calculator?
This Pythagorean Theorem Calculator solves for a single MISSING side of a right triangle when you already know the other two sides. The Triangle Calculator instead takes three given side lengths for any triangle (not necessarily a right triangle) and validates whether they form a real triangle, then computes its perimeter, area, and side classification (equilateral, isosceles, or scalene). Use this calculator when you're missing one right-triangle side; use the Triangle Calculator when you have all three sides and want to check validity, area, or classification.
Why do I need to choose which side I'm solving for?
The Pythagorean theorem relates three sides, but the two you already know and the one you're missing determine which rearrangement of a² + b² = c² actually applies. Solving for the hypotenuse uses c = √(a² + b²); solving for a leg instead subtracts inside the square root (a = √(c² − b²), or b = √(c² − a²)). Selecting a mode tells the calculator which two sides you're providing and which formula to apply.
Why does the calculator reject some hypotenuse/leg combinations?
In any right triangle, the hypotenuse is always the longest side — it's opposite the 90-degree angle. When solving for a leg, this calculator requires your entered hypotenuse value to be strictly greater than your entered known leg. If it isn't (for example, entering a "hypotenuse" of 4 with a known leg of 5), those two numbers cannot describe a real right triangle, so the calculator reports the problem clearly instead of taking the square root of a negative number, which would be mathematically meaningless.
Can the known sides be decimal (non-integer) values?
Yes. The Pythagorean theorem works identically for decimal side lengths — for example, legs of 1.5 and 2 produce a hypotenuse of exactly 2.5 (1.5² + 2² = 2.25 + 4 = 6.25, and √6.25 = 2.5). Any positive, finite number is accepted for either known side.
Why do both known-side fields say their meaning depends on the mode?
This calculator uses two generic fields — "Known Side 1" and "Known Side 2" — for all three modes rather than a separate set of labeled fields per mode, so switching modes doesn't require re-learning a new form layout. Each field's helper text spells out exactly what it means in every mode, and the "Formula Used" result restates which formula and which sides produced your answer.