Perimeter of a Rhombus – Definition, Formula, Examples, Facts, FAQs

The perimeter of a rhombus is the total length of its outer boundary. It is measured in linear units, such as feet, inches, yards, etc.

Perimeter of a rhombus $= 4 \times side$

A rhombus is a quadrilateral (a four-sided polygon) with the following properties:

• All sides are equal.
• Opposite angles are congruent.
• Adjacent angles are supplementary.
• Opposite sides are parallel.
• The altitude is a distance at right angles between two parallel sides.
• The diagonals of a rhombus bisect each other at the right angles.
• Each diagonal bisects the vertex angle. 

Its properties make it a unique and important geometric shape. Let’s explore the perimeter of rhombus in detail.

The image above shows a rhombus ABCD, along with its important properties.

So, what is the perimeter of the rhombus ABCD? How do you find the perimeter of a rhombus? Let’s discuss formulas and methods in detail.

Perimeter of a Rhombus: Formulas

Let’s discuss the formulas for the perimeter of a rhombus with two cases.

• When the side length is known

Perimeter of a rhombus with side “a” $= 4 \times a$

• When the length of the diagonals is known

Perimeter of a rhombus with diagonals p and $q = 2\sqrt{p^{2} + q^{2}}$

How to Find the Perimeter of a Rhombus

To find the perimeter of a rhombus, first note down the known dimensions. Make sure that all the measurements are in the same unit.

If the side length of a rhombus is known, use the first formula $P = 4\times$ side length. 

If the lengths of the two diagonals of a rhombus are known $(say\; d_{1}$ and $d_{2})$, use the formula $= 2\sqrt{d_{1}^{2} + d_{2}^{2}}$

Perimeter of a Rhombus Using the Side Lengths

The perimeter of a rhombus is defined as the total distance around the outline of the shape. It is the sum of the lengths of all four sides. Since all sides of a rhombus are equal, the formula for the perimeter is simply four times the length of one side.

Perimeter of a rhombus with side “x” $= x + x + x + x$

Perimeter $= 4\times $ Side Length $= 4x$

Example: Find the perimeter of a rhombus with a side length of 8 inches.

Perimeter $= 4\times$ Side Length

Perimeter $= 4 \times 8$ inches

Perimeter $= 32$ inches

So, the perimeter of the rhombus is 32 inches.

Perimeter of a Rhombus Using Diagonals

When the length of the diagonals of a rhombus is given, we first find the side length of the rhombus using the Pythagoras’ theorem. 

The properties of rhombus we use here are 

• A rhombus is divided into 4 congruent right triangles by its two diagonals.
• The diagonals bisect each other at right angles.

Let us consider a rhombus ABCD with diagonals p and q and with side length “a.”

Consider the triangle AOD. 

Since the diagonals bisect at right angles, we can write 

AO $= \frac{p}{2}$ and OD $= \frac{q}{2}$. 

On applying the Pythagoras’ theorem for triangle AOD, we get

$a^{2} = \frac{p^{2}}{4} + \frac{q^{2}}{4}$

$a = \sqrt{\frac{p^{2} + q{2}}{2}}$

Perimeter of a rhombus $= 4a =4 \times \sqrt{\frac{p^{2} + q^{2}}{2}} = 2\sqrt{p^{2} + q^{2}}$

Perimeter of a Rhombus Using One Diagonal and One Angle

Suppose that you know the length of one of the two diagonals and one interior angle. Now, to find the perimeter of the rhombus, you first need to find the length of the side using the different properties of a rhombus. Let’s understand this with an example. 

Example: Find the perimeter of the rhombus ABCD given below.

Given: $\angle A = 60^{\circ}$ and $AC = 20$

Each diagonal bisects the vertex angle. 

Thus, $\angle BAE = \frac{1}{2} \angle BAD = 30^{\circ}$

The diagonals of a rhombus bisect each other at right angles.

Thus, $\angle AEB = 90^{\circ}$ and $AE = \frac{1}{2}\times AC = 10$

In triangle AEB, $cos(\angle BAE) = cos(30^{\circ})= \frac{\sqrt{3}}{2}$

Also, $cos(\angle BAE) = \frac{AE}{AB}$

Thus,  $\frac{\sqrt{3}}{2} = \frac{10}{AB}$

$AB = \frac{20}{\sqrt{3}}$ …side of rhombus ABCD

Thus, perimeter of rhombus $ABCD = 4 \times side = 4 \times \frac{20}{\sqrt{3}} = \frac{80}{\sqrt{3}} \approx 46$ units

Conclusion

In this article, we learned how to find the perimeter of a rhombus and also the different methods to calculate the perimeter of a rhombus. Let’s use these concepts and move ahead to solve a few examples and practice problems.

Solved Examples on the Perimeter of a Rhombus

1. What is the perimeter of a rhombus with each side length equal to 10 inches?

Solution: 

Side of rhombus $= a = 10$ inches

Perimeter of rhombus $= 4a = 4 \times  8 = 32$ inches

2. Find the side of a rhombus if the perimeter is 36 feet.

Solution:  

Perimeter of a rhombus $= 36$ feet

Let the side of the rhombus be “a” feet.

$4a = 36$ feet

$a = \frac{36}{4} = 9$ feet

3. If the diagonals of the rhombus are 6 and 8 inches, then what is the perimeter of the rhombus?

Solution:  

Let $p = 6$ inches and $q = 8$ inches

Perimeter of a rhombus with diagonals p and $q = 2\sqrt{p^{2} + q^{2}}$

Perimeter of the rhombus $= 2\sqrt{6^{2} + 8^{2}}$

         $=2 \times 10$

         $=20$ inches

4. ABCD is a rhombus with sides 15 inches. What is the perimeter of the rhombus ABCD?

Solution:  

Side of rhombus $= a = 15$ inches

Perimeter of rhombus $= 4a = 4 \times 15 = 60$ inches

5. A field is in the shape of a rhombus with sides 13 feet. Find the length of the boundary of the field.

Solution:   

Length of the boundary of the field = Perimeter of the rhombus

Side of rhombus shaped field $= a = 13$ feet

Perimeter of the field $= 4a = 4 \times 13 = 42$ feet

Length of the boundary of the field $= 42$ feet

Practice Problems on Perimeter of a Rhombus

Frequently Asked Questions on Perimeter of a Rhombus