Pythagorean Theorem Calculator
Solve for a Missing Right-Triangle Side
Enter any two of the three sides of a right triangle and leave the third blank to solve it using a² + b² = c².
Understanding the Pythagorean Theorem
One of the oldest and most useful results in mathematics, the Pythagorean theorem links the three sides of any right triangle with a single elegant equation.
A right triangle has two legs (a and b) that meet at the 90° angle, and a hypotenuse (c) — always the longest side — opposite that angle.
Because the theorem relates all three sides through one equation, knowing any two of them is always enough to find the third — no angles required.
The Formula
Solving for a leg instead of the hypotenuse simply rearranges the formula: a = √(c² − b²) or b = √(c² − a²).
A Narrowly-Scoped Tool by Design
This calculator does exactly one job: solve a² + b² = c² for a missing side. If you need the full picture of a right triangle — both acute angles, area, and perimeter — use the Right Triangle Calculator. If your triangle doesn't have a right angle at all, the theorem doesn't apply directly; instead use the Triangle Calculator, which solves general triangles with the Law of Cosines and Law of Sines.
The theorem only holds for triangles containing a 90° angle.
The hypotenuse is opposite the right angle and never the shortest side.
Any two known sides are enough to solve for the third.
Key Takeaways
- a² + b² = c² only applies to right triangles — triangles with one 90° angle.
- The hypotenuse is always the longest side, so it can never be shorter than either leg.
- Need angles, area, or perimeter too? The Right Triangle Calculator solves the whole triangle, not just one side.
Frequently Asked Questions
- Enter any two of the three side lengths — leg
a, legb, or hypotenusec. - Leave the remaining field blank.
- Click "Calculate" to solve for the missing side.
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². It only applies to right triangles — triangles that contain a 90° angle.
The hypotenuse is always the longest side of a right triangle, and it is always the side directly opposite the 90° angle. The two shorter sides that form the right angle are called the legs (a and b).
This calculator solves specifically for one unknown side of a right triangle. Two known sides are enough to determine the third using a² + b² = c², so the calculator expects exactly one blank field to know which side to solve for. If you already know all three sides, there's nothing to solve — you can verify the triangle is a right triangle by checking that a² + b² = c² holds true.
That combination is mathematically impossible — the hypotenuse is always the longest side of a right triangle, so it can never be shorter than (or equal to) either leg. If you see this error, double-check which value belongs in the hypotenuse (c) field.
No — this tool is intentionally scoped to just the three side lengths of a right triangle. If you also need the two acute angles, area, and perimeter, use the Right Triangle Calculator instead, which fully solves a right triangle from any two known values, including an angle.