# Hypotenuse of a Triangle – Definition, Formulas

Area of an Isosceles Right Triangle with hypotenuse h
Area of an Isosceles Right Triangle with hypotenuse h

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 = √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.

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.

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.

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.

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.

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

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

= √2×2

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

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.

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