Pythagorean theorem calculator: solve for any right triangle side from a, b, or c with the steps shown. Free, private, runs in your browser, no signup.
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Pythagorean theorem calculator: solve for any right triangle side from a, b, or c with the steps shown. Free, private, runs in your browser, no signup.
Legs 9 and 12 give hypotenuse c = 15 exactly.
Transparent methodology
How this calculator works
Reviewed August 2026 · BoringToolsKit Editorial Team
Formula
Hypotenuse c = sqrt(a^2 + b^2); legs: a = sqrt(c^2 - b^2).
Worked example
Legs 3 and 4: hypotenuse 5, the classic 3-4-5 right triangle.
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Worked example: 9 and 12 make 15
Say you know two legs: a = 9 and b = 12, and you want the hypotenuse c. The formula squares each leg first: 9 squared is 81 and 12 squared is 144. Add them and you get 225. The hypotenuse is the square root of 225, which is exactly 15, so the calculator returns Hypotenuse (c) = 15 with no rounding needed. This is one of the classic Pythagorean triples, and triples like (3, 4, 5), (5, 12, 13), and (8, 15, 17) are worth memorizing because they let you sanity-check any calculator instantly.
The reverse direction works the same way with subtraction. Suppose the hypotenuse is 13 and one leg is 5. Square both: 169 and 25. Subtract to get 144, take the square root, and the missing leg is 12. Pick the 'a' or 'b' option, enter 13 as Value 1 and 5 as Value 2, and the tool reproduces that arithmetic. Note the order rule built into the calculator: in leg mode, Value 1 is treated as the hypotenuse, so entering a leg value there that is smaller than Value 2 produces a negative under the root and the check-inputs message instead of a number.
Not every pair lands on a whole number. Two legs of 1 give sqrt(2) = 1.4142 (shown to 4 decimal places), and legs of 7 and 4 give sqrt(65) = 8.0623. Those decimals are truncated for display, so a triple like legs 10 and 20 shows hypotenuse 22.3607 rather than the exact root of 500.
Where the formula stops being enough
The calculator assumes a true right angle. If the corner you measured is off by even a few degrees, the answer drifts. A triangle with legs 6 and 8 has a hypotenuse of 10 at exactly 90 degrees, but at 85 degrees the true third side is about 9.7, an error of 0.3, and at 80 degrees it is about 9.35. For layout work like checking a deck corner or a tile row, use the 3-4-5 method with real multiples (30-40-50 feet, for example) because scaling up makes small angle errors easier to spot.
Two edge cases produce the built-in error messages rather than a number: negative inputs, and leg mode where Value 1 is not larger than Value 2. A third silent trap is the degenerate result: legs of 0 and 5 return a hypotenuse of 5, which is mathematically consistent but describes a line, not a triangle. Zero-length legs are almost always a typo or a field left blank.
Also remember the result is a length, not a proof of squareness. Builders combine it with a diagonal check: a rectangle that measures 12 by 5 feet should have diagonals of exactly 13 feet (sqrt(144 + 25) = 13). If your tape reads 13 feet 1 inch on one diagonal, the corners are not square and the Pythagorean number has told you something is off, not what to cut.
