# Hypotenuse of a Triangle

## What is the Hypotenuse of a Triangle

A hypotenuse is the longest side of a right triangle. It is the side opposite the right angle (90°). The word ‘hypotenuse’ came from the Greek word ‘hypoteinousa’, meaning ‘stretching under’, where ‘hypo’ means ‘under’, and ‘teinein’ means ‘to stretch’.

## Formulas

### How to Find the Hypotenuse of a Right Triangle

a) When Base and Height are Given

To calculate the hypotenuse of a right or right-angled triangle when its corresponding base and height are known, we use the given formula.

Derivation

By Pythagorean Theorem,

(Hypotenuse)2 = (Base)2 + (Height)2

Hypotenuse = √(Base)2 + (Height)2

Thus, mathematically, hypotenuse is the sum of the square of base and height of a right triangle.

The above formula is also written as,

c = √a2 + b2, here c = hypotenuse, a = height, b = base

Let us solve some problems to understand the concept better.

Problem: Finding the hypotenuse of a right triangle, when the BASE and the HEIGHT are known.

What is the length of the hypotenuse of a right triangle with base 8m and height 6m.

Solution:

As we know,
c = √a2 + b2, here a = 6m, b = 8m
= √(6)2 + (8)2
= √36 + 64 = √100 = 10m

b) When Length of a Side and its Opposite Angle are Given

To find the hypotenuse of a right triangle when the length of a side and its opposite angle are known, we use the given formula, which is called the Law of sines.

Given as,

c = a/sin α = b/sin β, here c = hypotenuse, a = height, b = base, α = angle formed between hypotenuse and base, β = angle formed between hypotenuse and height

Let us solve some problems to understand the concept better.

Problem: Finding the hypotenuse of a right triangle, when the LENGTH OF A SIDE and its OPPOSITE ANGLE is known.

Find the length of hypotenuse in the given right triangle.

Solution:

Here, we will use the Law of sines formula,
c = a/sin α, here a = 12, α = 30°
= 12/ sin 30° = 12 x 2 = 24 units

Solve the length of hypotenuse in the given right triangle.

Solution:

Using the Law of sines formula,
c = b/sin β, b = 4, β = 60°
= 4/ sin 60° = 8/√3 units

c) When the Area and Either Height or Base are Known

To determine the hypotenuse of a right triangle when the height or base is known, we use the Pythagorean Theorem to derive the formula as shown below:

As we know from the Pythagorean Theorem

c = √(a)2 + (b)2…..(1), here c = hypotenuse, a = height, b = base

Again,

Area of right triangle (A) = a x b/2

b = area x 2/a …… (2)

a = area x 2/b …… (3)

Putting (2) in (1) we get,

c = √(a2 + (area x 2/a)2)

Similarly,

Putting (3) in (1) we get,

c = √(b2 + (area x 2/b)2)

Problem: Finding the hypotenuse of a right triangle, when the AREA and one SIDE are known.

What is the length of the hypotenuse of a right triangle with area 20m2 and height 6m.

Solution:

As we know,
c = √(a2 + (area x 2/a)2), here area = 20m2, a = 6m
= √62 + (20 x 2/6)2)
=√80.35 = 8.96 m

What is the length of the hypotenuse of a right triangle with area 14cm2 and base 9cm.

Solution:

As we know,
c = √(b2 + (area x 2/b)2), here area = 14cm2, b = 9cm
= √92 + (14 x 2/9)2)
= √45.67 = 6.75 m

### How to Find the Hypotenuse of a Right Isosceles Triangle

To derive the formula for finding the hypotenuse of a right isosceles triangle we use the Pythagorean Theorem.

As we know,

c = √a2 + b2

Let the length of the two equal sides be x, such that (a = b = x)

Then,

c =√x2 + x2

= √2x2

What is the length of the hypotenuse of a right isosceles triangle with two equal sides measuring 5.5 cm each.

Solution:

As we know,
c = √2x, here x = 5.5
= √2 x 5.5 = 7.77 cm

Find the measure of the length of the hypotenuse of a 45-45-90 triangle with one of the two equal sides measuring 9 cm.

Solution:

As a 45-45-90 triangle is a right isosceles triangle, we can apply the formula of right isosceles triangle for calculation of area
As we know,
c = √2x, here x = 9 cm
=√2 x 9 = 12.72 cm