Pythagorean Theorem Calculator

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a^2 + b^2 = c^2. For example, a triangle with sides 3 and 4 has a hypotenuse of 5 (9 + 16 = 25, sqrt(25) = 5).

Enter the two shorter sides (legs a and b). The calculator computes c = sqrt(a^2 + b^2). For legs of 6 and 8: c = sqrt(36 + 64) = sqrt(100) = 10. The result is shown as both an exact value (simplified radical when applicable) and a decimal approximation.

If you know the hypotenuse (c) and one leg (a), the other leg is b = sqrt(c^2 - a^2). For hypotenuse 13 and leg 5: b = sqrt(169 - 25) = sqrt(144) = 12. The calculator automatically determines which formula to use based on which values you enter.

Pythagorean triples are sets of three whole numbers that satisfy a^2 + b^2 = c^2. Common examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25), and any multiple of these (6,8,10 is 2x the 3,4,5 triple). The calculator identifies when your result forms a Pythagorean triple.

Yes — the Pythagorean theorem only applies to right triangles (triangles with one 90-degree angle). For non-right triangles, use the Law of Cosines (c^2 = a^2 + b^2 - 2ab*cos(C)), which is available in our Triangle Calculator. The Pythagorean theorem is actually a special case of the Law of Cosines where C = 90 degrees.