Area of Rhombus: Formula, Derivation and Examples

Rhombus is a parallelogram in which all four sides are equal and opposite pairs of lines are congruent. The opposite angles in a rhombus are equal.

Let’s learn the formulas of the area of rhombus, their derivation and examples in detail.

Area of Rhombus

Area of rhombus is defined as the space enclosed by the Rhombus in the 2-D plane. It depends on the dimensions of the rhombus.

It is measured in square units, such as square meters, square centimeters, etc.

Note– Rhombus often gets confused with square but rhombus is very different from the square.

Area of Rhombus Formula

Area of the rhombus can be found using various methods some of them are listed in the table below

Area of Rhombus Calculation using Formula
If Base and Height are givenA = b Ã— h
If Diagonals are givenA = Â½ Ã— D Ã— d
If Base and Interior angle is givenA = b2 Ã— Sin(a)

Where,

D = length of first diagonalÂ
d = length of second diagonal
b = length of side of rhombus
h = height of rhombus
a = measure of an interior angle

Illustration of Area Of Rhombus Formula

Area of Rhombus Formula Derivation

Let us consider a rhombus ABCD with O as the point of intersection of two diagonals AC and BD.

Derivation of Area of Rhombus

The area of rhombus will be

Area = 4 Ã— area ofÂ â–³AOB

Â  Â  Â  Â  = 4 Ã— (1/2) Ã— AO Ã— OB sq.units

Â  Â  Â  Â  = 4 Ã— (1/2) Ã— (1/2) d1 Ã— (1/2) d2 sq.unit

Â  Â  Â  Â  = 4 Ã— (1/8) d1 Ã— d2

Â  Â  Â  Â  Â  Â = 1/2 d1 Ã— d2

Therefore, the area of a rhombus is A = 1/2 d1 Ã— d2.

How to Find Area of Rhombus

The area of the rhombus can be calculated by three different methods by using diagonal, using base and height, and using trigonometry.

These are the three important methods for finding area of Rhombus:

1. Area of Rhombus when DiagonalsÂ are given
2. Area of Rhombus using Base and Height
3. Area of Rhombus using Trigonometric Ratios

Let’s discuss all these methods in detail.

Area of Rhombus with DiagonalsÂ

Area = (d1 Ã— d2)/2 sq. units

Where,

d1 is the length of diagonal 1

d2 Â is the length of diagonal 2

Let’s try to understand this formula with the help of an example.

Example 1: Find the area of a rhombus having diagonals 16 m and 18 m.

Solution:

Diagonal 1, d1 = 16 m

Diagonal 2, d2 = 18 m

Area of a rhombus, A = (d1 Ã— d2) / 2

= (16 Ã— 18) / 2

= 288 / 2

= 144 m2

Thus, the area of the rhombus is 144 m2

Area of Rhombus using Base and Height

Area of a Rhombus = b Ã— h sq units

Where,

b is the length of any side of the rhombus

h is the height of the rhombus

Example 2: Find the area of a rhombus having base of 12 m and height is 16 m.

Solution:

Base, b = 12 m

Height, h = 16 m

Area, A = b Ã— h

Â  Â  Â  Â  Â  Â  = 12 Ã— 16 m2

Â  Â  Â  Â  Â A = 192 m2

Thus, the area of the rhombus is 192 m2

Area of Rhombus using Trigonometric Ratios

Area of a Rhombus = b2 Ã— sin(A) sq. units

Where,

b is the length of any side of the rhombus

A is a measure of any interior angle

Example 3: Find the area of a rhombus if the length of its side is 12 m and one of its angles A is 60Â°

Solution:

Side = s = 12 m

Angle A = 60Â°

Area = s2 Ã— sin (60Â°)

A = 144 Ã— âˆš3/2

A = 72âˆš3 m2

Area of Rhombus Examples

Now let us solve some examples on the formulas we learnt on the area of rhombus.

Example 1: Calculate the area of a rhombus (using base and height) if its base is 5cm and height is 3cm.

Solution:

Given,

Base (b) = 5cm

height of rhombus(h) = 3cm

Now,’

Area of the rhombus(A) = b Ã— h

Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  = 5 Ã— 3

Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  = 15cm2

Example 2: Calculate the area of a rhombus (using diagonal) having diagonals equal to 4cm and 3cm.

Solution:

Given,

Length of diagonal 1 (d1) = 4cm

Length of diagonal 2 (d2) = 3cm

Now,

Area of Rhombus (A) = 1/2 d1 Ã— d2

Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â = 4 x3/2 = 6cm2

Example 3: Calculate the area of the rhombus (using trigonometry) if its side is 8cm and one of its angles A is 30 degrees.

Solution:

Side of the rhombus (b) = 8cm

angle (a) = 30 degrees

Now,

Area of the rhombus(A) = b2 Ã— sin(a)

Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  = (8) Ã— sin(30)

Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  = 64 Ã— 1/2 = 32 cm2

Example 4: Calculate the base of a rhombus if its area is Â 25cm2 and height is 10cm.

Solution:

Given,

Area = 25 cm2

height of rhombus(h) = 10 cm

Now,

Area of the rhombus(A) = b Ã— h

Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  25 Â = b Ã— 10

Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  = 2.5 cm

Area of Rhombus in Maths -FAQs

1. What do we mean by Rhombus?

A rhombus is a type of quadrilateral whose opposite sides are parallel and equal. Also, the opposite angles of a rhombus are equal and the diagonals bisect each other at right angles.

2. What is the Formula of Area of a Rhombus.

For finding the area of a rhombus, the given formula is used:

A = Â½ Ã— d1 Ã— d2

where, d1 and d2 are diagonals of rhombus

3. How to calculate the perimeter of a rhombus?

The perimeter of a rhombus can be calculated by the formula

P= 4b units

where b is a side of the rhombus.

4. How to find the Area of a Rhombus when the Side and Height are given?

The area of a rhombus its height and side are given is calculated using

A = Base Ã— Height sq units

5. How to find the area of rhombus with diagonals?

The area (A) of a rhombus when the lengths of its diagonals (d1 and d2) is given by the following formula:

A = (1/2) x d1 x d2

where,

A represents the area of the rhombus

d1 and d2 represent the lengths of the two diagonals.

6. What is the Area of Rhombus formula without diagonals?

When diagonals are not given, Area of a Rhombus can be calculated by the following formula,

Area of a Rhombus = b2 Ã— sin(A) sq. units

where,

b is the length of any side of the rhombus

A is a measure of any interior angle

7. Is the area of a rhombus the same as the area of a square?

No, the area of a rhombus is not the same as the area of a square.

8. What is the difference between the area of a rhombus and the area of a square?

The area of a rhombus is equal to the half the product of its diagonals, whereas the area of a square is calculated as the square of the length of its side. This shows their different geometric properties despite both being quadrilaterals.

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