What is the Pythagorean theorem calculator?
A Pythagorean theorem calculator finds the missing side of a right triangle using the formula a² + b² = c². In a right triangle, the two shorter sides (legs) are a and b, and the longest side (hypotenuse), opposite the right angle, is c. You enter two of the three sides and leave the third empty; the calculator computes the missing value. No sign-up or download is required; the tool runs in your browser and is useful for geometry, construction, and any situation where you need a missing side of a right triangle.
The Pythagorean theorem is one of the most famous results in mathematics: in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. This calculator applies that relationship so you can find the hypotenuse from both legs, or either leg from the hypotenuse and the other leg. Use it for homework, carpentry, design, or any right-triangle problem.
How to use this Pythagorean theorem calculator
Enter values for two of the three sides (leg A, leg B, hypotenuse C) and leave the third field empty. To find the hypotenuse: enter both legs (e.g. 3 and 4) and leave hypotenuse empty—the calculator gives c = √(a² + b²) = 5. To find a leg: enter the other leg and the hypotenuse (e.g. leg 3 and hypotenuse 5) and leave the missing leg empty—the calculator gives the missing leg. The hypotenuse must always be longer than either leg; if you enter a hypotenuse shorter than or equal to a leg, the calculator will show an error. Use positive numbers. Results appear as soon as exactly two sides are entered and one is empty.
The Pythagorean theorem formula
The Pythagorean theorem states that in a right triangle, a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse. So the hypotenuse is c = √(a² + b²). To find a leg when you know the other leg and the hypotenuse, rearrange: a² = c² − b², so a = √(c² − b²), and similarly b = √(c² − a²). For example, legs 3 and 4 give hypotenuse √(9 + 16) = 5. Leg 3 and hypotenuse 5 give the other leg √(25 − 9) = 4. This calculator applies these formulas automatically when you leave one side empty.
Why does a² + b² = c²?
The theorem can be proved in many ways. One classic proof uses four copies of the right triangle arranged in a square: the side length of the outer square is a + b, and the area of the outer square equals the area of the four triangles plus the area of the inner square (whose side is c). Algebra shows that (a + b)² = 4 × (½ab) + c², which simplifies to a² + b² = c². Another proof uses similar triangles and the fact that the altitude to the hypotenuse divides the triangle into two triangles similar to the original. Regardless of the proof, the relationship holds for every right triangle, and this calculator uses it to find the missing side.
Right triangles and the hypotenuse
A right triangle has one 90° angle. The side opposite that angle is the hypotenuse and is always the longest side. The two other sides are the legs. So when you know two sides, the missing side is uniquely determined: if you know both legs, the hypotenuse is √(a² + b²); if you know the hypotenuse and one leg, the other leg is √(c² − leg²). The calculator expects the hypotenuse in the “Hypotenuse C” field. If you mistakenly put the longest side in a leg field, the formula c² − a² or c² − b² could be negative, and the calculator will show an error because the hypotenuse must be greater than each leg.
Common Pythagorean triples
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². Common triples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Multiples of these are also triples: (6, 8, 10), (9, 12, 15), etc. You can use this calculator to verify: enter 3 and 4 and leave c empty to get 5; enter 5 and 13 and leave one leg empty to get 12. These triples are useful in construction and layout when you need exact right angles (e.g. 3-4-5 for squaring a corner).
Applications of the Pythagorean theorem
The Pythagorean theorem is used in construction (squaring corners, roof pitch), navigation (distance between points), physics (resultant forces, vectors), and design (diagonals, screen sizes). Whenever you have a right triangle and know two sides, the third is given by the theorem. This Pythagorean theorem calculator gives you the missing side quickly and accurately. Always use the same units for all sides; the result will be in that same unit. For legs in different units, convert first.
Summary
This Pythagorean theorem calculator finds the missing side of a right triangle using a² + b² = c². Enter two sides (legs a and b, or one leg and hypotenuse c) and leave one field empty to get the missing value. The tool is free, runs in the browser, and requires no account. Use it for homework, construction, or any right-triangle problem. Remember: c is the hypotenuse (longest side, opposite the right angle), and it must be greater than either leg.