# 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](/content/math/geometry/right-angled-triangle/index.html) 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](/content/math/geometry/isosceles-right-triangle/index.html), 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](https://en.wikipedia.org/wiki/Pythagorean_theorem) 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.