Pythagorean Theorem Calculator

The hypotenuse is the side directly across from the right angle (the 90° corner). It is always the longest side of any right triangle. In diagrams it is usually labeled $c$. If you see a small square drawn in a corner, the side opposite that square is the hypotenuse.

A leg cannot be longer than or equal to the hypotenuse, because $a^2 + b^2 = c^2$ requires $c > a$ and $c > b$. For example, a hypotenuse of 4 with a leg of 5 is impossible — the leg is already longer than the hypotenuse. The calculator detects this and shows an error instead of a mathematically nonsensical result.

No — $a^2 + b^2 = c^2$ holds only when the angle between the two legs is exactly 90°. For acute triangles $a^2 + b^2 > c^2$, and for obtuse triangles $a^2 + b^2 < c^2$. These inequalities are used to classify triangles. To solve an arbitrary (non-right) triangle you need the Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$.

Once all three sides are known, the acute angles follow from inverse trigonometry. Angle $A$ (opposite leg $a$) is $\arctan(a/b)$, and angle $B$ (opposite leg $b$) is $\arctan(b/a)$. Together with the 90° right angle, the three angles always sum to 180°.

A Pythagorean triple is a set of three positive integers ($a$, $b$, $c$) that satisfy the theorem exactly — no decimals. The most famous is 3-4-5. Others include 5-12-13, 8-15-17, and 7-24-25. Multiples of any triple (like 6-8-10 or 9-12-15) also work. Builders use the 3-4-5 triple to verify square corners quickly with just a tape measure.

This calculator focuses specifically on the Pythagorean theorem — solving for a missing side from two known sides. The right-triangle calculator on this site solves the triangle from two legs using both the theorem and trigonometry, outputting all angles and the hypotenuse in one step.