Hypotenuse - Definition, Formula, and Examples

TL;DR

The hypotenuse is the side of a right triangle that lies opposite the right angle, and it is always the longest side. You find it with the Pythagorean theorem, c²=a²+b²; this article covers the definition, the formula, six worked examples, and the mistakes students make most.

What Is A Hypotenuse?

A hypotenuse is the longest side of a right triangle - a triangle that contains one 90° angle - and it is the side directly opposite that right angle. The other two sides, which form the right angle between them, are called the legs (sometimes the base and the perpendicular).

Only right triangles have a hypotenuse. An equilateral or an obtuse triangle has three sides, but none of them earns the name, because there is no right angle for a side to sit opposite. The word itself comes from the Greek hypoteinousa, meaning "stretching under" — the side that stretches across from corner to corner.

Here is the fact that makes the hypotenuse special: the side opposite the largest angle in any triangle is the longest side. In a right triangle the 90° angle is the largest angle possible, because the other two must share the remaining 90°. So the side across from it wins the length contest every time.

The right-angled triangle is where this all lives, and the length of the hypotenuse is locked in by the two legs through one formula.

What Is The Hypotenuse Formula?

The hypotenuse length comes straight from the Pythagorean theorem:

c²=a²+b²

Solving for the hypotenuse c:

c=a²+b²

Examples of Hypotenuse

Six worked examples, easier to harder. The problem statement is bold; the steps are not.

Example 1

Find the hypotenuse of a right triangle with legs 3 and 4.

c=√(3² + 4²) c=√(9 + 16) c=√(25) c=5

Final answer: the hypotenuse is 5. This 3-4-5 set is the most famous Pythagorean triple and worth memorising.

Example 2

A student is told the two legs are 6 and 8 and writes the hypotenuse as 6 + 8 = 14. Is that right?

The tempting move is to add the legs directly, because the hypotenuse "goes further," so it should be bigger. Adding gives 14.

Check it against the theorem: c=√(6² + 8²) c=√(36 + 64) c=√(100) c=10

The real answer is 10, not 14. Adding the legs ignores that you square first, add, then take the root - the squaring is what keeps the hypotenuse shorter than the straight sum of the two legs.

Example 3

Find the hypotenuse when both legs are 5.

c=√(5² + 5²) c=√(25 + 25) c=√(50) c=5√2

Final answer: 5√2, about 7.07. A right triangle with two equal legs is an isosceles right triangle, and its hypotenuse is always a leg times √2.

Example 4

A ladder leans against a wall. Its foot is 5 m from the wall and it reaches 12 m up. How long is the ladder?

The ladder is the hypotenuse; the wall height and the ground distance are the legs. c=√(5² + 12²) c=√(25 + 144) c=√(169) c=13

Final answer: the ladder is 13 m long.

Example 5

The hypotenuse is 13 and one leg is 5. Find the other leg.

Now you rearrange the formula to solve for a leg: b=√(13² - 5²) b=√(169 - 25) b=√(144) b=12

Final answer: the missing leg is 12.

Example 6

Two towns sit at coordinates A(1,2) and B(4,6). How far apart are they in a straight line?

The straight-line gap is the hypotenuse of a right triangle whose legs are the horizontal and vertical separations.

Horizontal leg: 4−1=3 Vertical leg: 6−2=4

distance=√(3² + 4²) distance=√(9 + 16) distance=√(25) distance=5

Final answer: the towns are 5 units apart.

Why the Hypotenuse Earns Its Name

The hypotenuse is not just a vocabulary word - it is the reason right triangles run so much of applied geometry.

Ask why humans needed it, and the answer is old and practical:

• Surveying and building. Long before coordinates existed, builders needed a reliable way to make a true right angle in the field.
• Measuring the unreachable. You cannot lay a tape measure up the face of a cliff or across a river.
• The straight-line shortcut. Any time you ask "how far apart are two points," you are asking for a hypotenuse.

The idea is far older than its name. Clay tablets show that Babylonian mathematicians worked with these side relationships more than a thousand years before Pythagoras.

The Mistakes Students Make Most Often

Three failure modes cover almost every wrong hypotenuse answer.

Mistake 1: Adding the legs instead of using the theorem

Where it slips in: the very first time a student sees two leg lengths and is asked for the hypotenuse.

The correct way: square, add, then root: c=√(6² + 8²).

Mistake 2: Solving for a leg with addition instead of subtraction

Where it slips in: problems that give you the hypotenuse and one leg and ask for the other leg.

The correct way: when the hypotenuse is given, subtract: b=√(c² - a²).

Mistake 3: Calling any long side a hypotenuse

Where it slips in: triangles that are not right triangles at all.

The correct way: first confirm there is a 90° angle. No right angle means no hypotenuse.

Conclusion

• The hypotenuse is the side of a right triangle opposite the 90° angle, and it is always the longest side.
• Find it with the Pythagorean theorem: c=√(a² + b²).
• To find a missing leg when the hypotenuse is known, subtract: b=√(c² - a²).
• Never add the legs directly - square, add, then take the root.
• Only right triangles have a hypotenuse; always confirm the right angle first.