# Hypotenuse Formula

The hypotenuse formula is used in a right-angled triangle, to calculate the longest side that is called the hypotenuse. The hypotenuse is always the side that is opposite to the right angle in a right triangle. We use the Pythagoras theorem to find the hypotenuse, and thus the hypotenuse formula.

## What Is Hypotenuse Formula?

Hypotenuse formula or the Pythagoras theorem is defined as the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs). Let us understand the derivation of the hypotenuse formula with the help of the Pythagoras theorem.

### Pythagoras Theorem

Here,

- c = hypotenuse
- a = height (a leg)
- b = base (a leg)

From Pythagoras theorem, (hypotenuse)^{2} = (base)^{2} + (height)^{2}

If we take the square root on both sides, we get the formula of the hypotenuse.

**Hypotenuse formula = √((base) ^{2} + (height)^{2}) (or) c = **√

**(a**

^{2}+ b^{2})## Examples on Hypotenuse Formula

Let us solve some interesting problems using the hypotenuse formula.

**Example 1: **Using hypotenuse formula solve for the longest side of the given bread slice that is similar to a right-angle triangle. Its height is 13 units and its base is 5 units.

**Solution: **

To find: The longest side (hypotenuse) of the bread slice.

Its height = 13 units.

Its base = 5 units.

By using the hypotenuse formula,

hypotenuse formula = √ ((base)^{2} + (height ^{2})

=√(5^{2}+13^{2})

=√25+169

=√194

=13.98

**Answer: The longest side of the bread slice = 13.928 units.**

**Example 2:** The hypotenuse of a right triangle is 5 units. Its base is 3 units. Find its height using the hypotenuse formula.

**Solution: **

To find: The height of the right triangle.

Let us assume it to be 'h'

Its hypotenuse = 5 units.

Its base = 3 units.

By using the hypotenuse formula,

hypotenuse^{2} = (base)^{2} + (height)^{2}

5^{2 }= 3^{2 }+ h^{2}

25 = 9 + h^{2}

25−9 = h^{2}

16 = h^{2}

4 = h

**Answer: The height of the right triangle = 4 units.**

**Example 3: **Using the hypotenuse formula, solve for the longest side of the given bread slice that is similar to a right-angle triangle. Its height is 14 units and its base is 8 units.

**Solution:**

To find: The longest side (hypotenuse) of the bread slice.

Its height = 14 units.

Its base = 8 units.

By using the hypotenuse formula,

hypotenuse formula = √((base)^{2} + (height ^{2})

=√(8^{2}+14^{2})

=√64+196

=√260

=16.124

**Answer: The longest side of the bread slice = 16.124 units.**

## FAQs on Hypotenuse Formula

### How do you Find the Hypotenuse of a Triangle Using the Hypotenuse Formula?

By using the Pythagorean theorem (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}**, **we can calculate the hypotenuse. If the values of the other two sides are known, the hypotenuse can be easily calculated with this formula. It is also known as the hypotenuse formula.

### How Do you Find the Longest Side of a Triangle Using the Hypotenuse Formula?

The hypotenuse is termed as the longest side of a right-angled triangle. To find the longest side we use the hypotenuse formula that can be easily driven from the Pythagoras theorem, (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}.

Hypotenuse formula = √((base)^{2} + (height)^{2}) (or) c = √(a^{2} + b^{2}).

### How Do You Find the Hypotenuse Formula Using the Pythagoras Theorem?

The Pythagoras theorem is represented by the formula (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}. The longest side of the right triangle is the hypotenuse and the other two sides are the base and the altitude of the triangle. Thus the hypotenuse formula is expressed as:

Hypotenuse formula = √((base)^{2} + (height)^{2})