Hypotenuse Calculator

The hypotenuse is the side opposite the right angle in a right triangle, and it is always the longest side. Everything below assumes the triangle genuinely has a $90°$ angle — none of it applies otherwise.

From two legs (Pythagorean theorem)

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

From a leg and an acute angle

$c = \frac{\text{opposite}}{\sin\theta} = \frac{\text{adjacent}}{\cos\theta}$

From the hypotenuse back to a leg (the hypotenuse-leg case)

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

Notice the subtraction: going from the hypotenuse reverses the operation. If $c^2 - b^2$ comes out negative, the numbers do not describe a real triangle — you have mislabelled which side is the hypotenuse.

Two triangles appear often enough that the arithmetic is worth skipping.

45-45-90 (isosceles right triangle)

The legs are equal and

$c = a\sqrt{2}, \qquad a = \frac{c}{\sqrt{2}} = \frac{c\sqrt{2}}{2}$

Side ratio $1 : 1 : \sqrt{2}$.

30-60-90

With the short leg (opposite $30°$) called $s$:

$\text{short leg} = s, \quad \text{long leg} = s\sqrt{3}, \quad \text{hypotenuse} = 2s$

Side ratio $1 : \sqrt{3} : 2$. The hypotenuse is exactly twice the short leg — never twice the long leg.

Choosing a method

Common integer triples worth recognising: $3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$, $8\text{-}15\text{-}17$, $7\text{-}24\text{-}25$, and any multiple of them.

• Adding before squaring: $c \neq a + b$. Square each leg first, add, then take the square root once at the end.
• Treating a leg as the hypotenuse: the hypotenuse sits opposite the right angle and must be longer than either leg. If your answer is smaller than a given side, you solved the wrong case.
• Using the theorem on a non-right triangle: without a $90°$ angle you need the Law of Cosines.
• Multiplying by $\sin$ when you should divide: with the opposite side known, $c = a/\sin\theta$. Multiplying shrinks the triangle instead of growing it.
• Mixing up the 30-60-90 legs: the $\sqrt{3}$ belongs to the long leg, opposite the $60°$ angle.