The Pythagorean Theorem & Its Converse

Finding the hypotenuse

When you know both legs, square each one, add the squares, then take the square root. With legs $6$ and $8$: $c^2 = 6^2 + 8^2 = 36 + 64 = 100$, so $c = 10$.

The most common mistake is adding the sides before squaring — $6 + 8 = 14$ squared is not the same as $6^2 + 8^2$. Square first, then add.

In the figure, the legs $a$ and $b$ meet at the right angle, and the hypotenuse $c$ is the side across from it.

Finding a leg

When you know the hypotenuse and one leg, the missing side is a leg — so you subtract instead of add. From $a^2 + b^2 = c^2$, the missing leg is $\sqrt{c^2 - a^2}$.

Before you compute anything, ask: is the missing side the hypotenuse or a leg? If it is across from the right angle, add the squares. If not, subtract the known leg's square from the hypotenuse's square. A quick sanity check: the hypotenuse must be the longest side, so a leg answer should come out smaller than the hypotenuse.

The converse: testing three sides

Take the three side lengths, call the longest one $c$, and compare $a^2 + b^2$ with $c^2$. If they are equal, the triangle is right. If $a^2 + b^2 > c^2$, the triangle is acute. If $a^2 + b^2 < c^2$, it is obtuse.

For example, $5$, $12$, $13$ gives $25 + 144 = 169 = 13^2$, so that triangle is right. But $6$, $8$, $11$ gives $36 + 64 = 100$ while $11^2 = 121$ — the sides fail the test, and since $c^2$ is bigger, the triangle is obtuse.

Worked examples

Try one yourself

Common questions