Table of Contents

Last modified on August 3rd, 2023

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’.

**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 = √a^{2} + b^{2}, 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 = √a^{2} + b^{2}, 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 = √(a^{2} + (area x 2/a)^{2})

Similarly,

Putting (3) in (1) we get,

c = √(b^{2} + (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 20m ^{2} and height 6m.**

Solution:

As we know,

c = √(a^{2} + (area x 2/a)^{2}), here area = 20m^{2}, a = 6m

= √6^{2 }+ (20 x 2/6)^{2})

=√80.35 = 8.96 m

**What is the length of the hypotenuse of a right triangle with area 14cm ^{2} and base 9cm.**

Solution:

As we know,

c = √(b^{2} + (area x 2/b)^{2}), here area = 14cm^{2}, b = 9cm

= √9^{2 }+ (14 x 2/9)^{2})

= √45.67 = 6.75 m

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

As we know,

c = √a^{2} + b^{2}

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

Then,

c =√x^{2} + x^{2}

= √2x^{2}

**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

Last modified on August 3rd, 2023