Hypotenuse

Which side it is

In a right triangle:

• the two sides meeting at the right angle are the legs
• the third side, opposite the right angle, is the hypotenuse

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

$c$ is always the hypotenuse. Putting a leg in its place gives a wrong answer every time.

It is always the longest

This is not a coincidence — it follows from the theorem. Since $a^2$ and $b^2$ are both positive, $c^2$ is larger than either, so $c$ is larger than either leg.

That gives a check that needs no arithmetic: a leg can never come out longer than the hypotenuse. An answer that does is wrong on inspection, and the usual cause is adding where the calculation should subtract.

You know	You want	Do
both legs	the hypotenuse	add the squares
the hypotenuse and a leg	the other leg	subtract the squares

Spotting it in a word problem

The hypotenuse is whichever length spans the two others — usually the sloping or diagonal one.

Situation	The hypotenuse
a ladder against a wall	the ladder
a diagonal across a rectangle	the diagonal
a ramp up to a step	the ramp

Naming it first is most of the work; the arithmetic follows. See the Pythagorean theorem.

Only in right triangles

A triangle without a right angle has no hypotenuse, and the theorem does not apply to it. The word and the formula arrive together.