Perimeter of Rhombus Formula

Last Updated : 12 Aug, 2024

In mensuration, the perimeter of a is defined as the sum of lengths of all the sides of the quadrilateral around the border. So perimeter of the rhombus is defined as the sum of all 4 sides of the rhombus.

Rhombus is a diamond-shaped quadrilateral whose all sides are equal but each angle inclined between these two sides is not equal. Since it is a quadrilateral it has four sides and all four sides are of equal length. It has the following properties.

Properties of Rhombus

All the sides are equal in length and opposite sides are parallel to each other.
Adjacent angles sum to 180 degrees and opposite angles remain the same.
The diagonals bisect each other perpendicularly and bisect the angles between the sides i.e vertex angles.
The Sum of all the angles in the Rhombus is 360 degrees.
The rhombus is a square if each vertex angle is equal to 90 degrees.

The shape of the Rhombus:

Perimeter of Rhombus using Side Lengths

Formula according to the definition:

Perimeter of Rhombus = 4×s

where

s is the side length of the rhombus.

Derivation:

Perimeter(P) = s + s + s + s = 4*s

Perimeter of Rhombus using Diagonal Lengths

Given horizontal diagonal length as a and vertical diagonal length as b then perimeter is given by:

P = 2 * √(a² + b²)

Derivation:

Since diagonals bisect each other at right angles each quadrant forms a right angled triangle and the lengths of the sides i.e base and height are a/2 and b/2 and side of Rhombus as s.

By applying Pythagoras Theorem:

a²/4 + b²/4 = s² (side)

s = (√(a² + b²))/2

P = 4 * s = 2 * √(a² + b²)

Sample Problems

Question 1: Find the perimeter of a rhombus whose side is 8 cm.

Solution:

Given that side s = 8 cm

Perimeter of Rhombus is given by : 4*s

So, Perimeter (P) = 4 * 8 cm = 32 cm

Question 2: Find the side length of a rhombus whose perimeter is given as 36cm.

Solution:

Given Perimeter(P) = 36 cm

P = 4 * s

=> s = P/4

So, s = 36/4 = 9cm

Question 3: Find the perimeter of the rhombus given the diagonal lengths are 6 cm and 8 cm respectively.

Solution:

When diagonal lengths are given :

Given a = 6 cm, b = 8cm

Perimeter(P) = 2* √(a² + b²) = 2* √(36 + 64) = 2 * 10 = 20 cm

Question 4: Find the length of horizontal diagonal given the side length as 13cm and vertical diagonal length as 24 cm.

Solution:

Since the diagonal bisect at right angles:

Given b = 24 cm and s = 13 cm, a = ?

side(s) is given as

s = (√(a² + b²))/2

2 * s = (√(a² + b²))

26 = (√(a² + 576))

On squaring both sides, 676 = a² + 576

=> a² = 100

=> a= 10cm

Question 5: Find the area of the rhombus whose diagonal are of lengths 24cm and 10 cm.

Solution:

Given a = 24 and b = 10cm

Area of the Rhombus is given by A = 1/2 * a * b

= 1/2 * 24 * 10

= 60 cm²

Question 6: Find the perimeter of a rhombus whose side is 2.5 cm.

Solution:

Given that side s = 8 cm

Perimeter of Rhombus is given by: 4*s

So, Perimeter (P) = 4 * (2.5) cm = 10 cm

Question 7: Find the side length of a rhombus whose perimeter is given as 48cm.

Solution:

Given Perimeter(P) = 48 cm

P = 4 * s

=> s = P/4

So, s = 48/4 = 12cm

Practice Problems on Perimeter of Rhombus Formula

Find the perimeter of a rhombus whose side is 9 cm.
Find the side length of a rhombus if its perimeter is 81 cm.
Find the perimeter of a rhombus given diagonal lengths of 3 cm and 4 cm.
If the side of a rhombus is 5 cm, what is its perimeter?
A rhombus has a side length of 7 cm. Calculate its perimeter.
Given diagonals of 12 cm and 16 cm, find the perimeter of the rhombus.
Find the length of the horizontal diagonal if the side length is 13 cm and the vertical diagonal is 24 cm.
Calculate the area of a rhombus with diagonals of 24 cm and 10 cm.
Find the perimeter of a rhombus whose side is 2.5 cm.
Determine the side length of a rhombus with a perimeter of 48 cm

Summary

The perimeter of a rhombus can be calculated using the length of its sides or its diagonals.The perimeter is simply four times the side length, or it can be derived using the lengths of the diagonals with the help of the Pythagorean theorem.