Pythagorean Theorem Calculator: Hypotenuse or Leg

The Pythagorean theorem

For any right triangle — one containing a 90° angle — the two shorter sides (the legs, $a$ and $b$) and the longest side (the hypotenuse, $c$, always opposite the right angle) satisfy:

$a^2 + b^2 = c^2$

Given any two of the three sides, this calculator solves for the third, two ways:

$c = \sqrt{a^2 + b^2} \qquad\qquad b = \sqrt{c^2 - a^2}$

The right triangle's area follows directly from the two legs: $A = \dfrac{ab}{2}$.

Solving for the hypotenuse

Given both legs, square each one, add the squares, and take the square root:

Example: legs a = 6, b = 8

$c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

$A = \frac{6 \times 8}{2} = 24$

Since 100 is a perfect square, the hypotenuse comes out exactly whole — no radical needed.

Example: legs a = 1, b = 1

$c = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.4142$

Here 2 has no perfect-square factor, so $\sqrt{2}$ is already in simplest form — the calculator shows both the exact radical and its decimal approximation. Notably, the area is still an exact $A = \frac{1 \times 1}{2} = 0.5$, even though the hypotenuse itself is irrational — area and hypotenuse are independent questions.

Solving for a missing leg

Given the hypotenuse and one leg, rearrange the theorem to isolate the other leg:

Example: hypotenuse c = 13, leg a = 5

$b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$

This is the well-known 5-12-13 triple. The area is $A = \frac{5 \times 12}{2} = 30$.

Pythagorean triples

A Pythagorean triple is a set of three positive integers satisfying $a^2+b^2=c^2$ exactly, so every side — and the hypotenuse — comes out whole. The smallest is:

Any scaled multiple of a known triple — like (6, 8, 10), twice (3, 4, 5) — is also a valid triple.

Common mistakes

• Mixing up the hypotenuse with a leg. The hypotenuse is always the longest side and always opposite the right angle — plugging a leg's value in for $c$ (or vice versa) gives a nonsensical result.
• Writing $c = a + b$. The theorem is about the SQUARES of the sides, not the sides themselves — legs 6 and 8 give a hypotenuse of 10, not 14.
• Applying it to a non-right triangle. $a^2+b^2=c^2$ only holds when the triangle has a genuine 90° angle. For any other triangle, use the Law of Cosines instead.

Where this shows up

• Distance between two points: the Distance Formula is this exact theorem applied to coordinates, treating the horizontal and vertical differences as the two legs.
• A square or rectangle's diagonal: the diagonal is the hypotenuse of a right triangle formed by two adjacent sides.
• Construction: builders check a corner is truly square using a 3-4-5 triangle — if three marks at those proportions form a closed triangle, the corner is exactly 90°.

The Pythagorean theorem is also the special case of the Law of Cosines when the included angle is exactly 90° (since $\cos 90° = 0$ makes the extra term vanish) — the same underlying relationship this calculator's Distance Formula and Simplify Radicals calculators build on.