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’.
Hypotenuse of a Triangle
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.
Hypotenuse of a Right Triangle 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.
Formula to Find Hypotenuse of a Right Triangle
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.
Hypotenuse of an Isosceles Right Triangle Formula
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
Find the length of hypotenuse in the given right triangle.
Solve the length of hypotenuse in the given right triangle.