Perimeter of Rhombus Formula

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.

• 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