Class 8 - Area Of Rhombus Using Diagonal Method

Rhombus is a quadrilateral having all sides of equal length. We can use the same triangulation method discussed above to find the area of a rhombus. Consider a rhombus BOAT with diagonals AB and OT. The diagonals of a rhombus are the perpendicular bisectors of each other.

Table of Contents 

• What is the Area of a Rhombus?
• Area of Rhombus Formula
• Derivation for Rhombus Area Formula
• How to Calculate the Area of a Rhombus?
• Method 1: Area of Rhombus Using Diagonals
• Method 2: Area of Rhombus Using Base and Height
• Method 3: Area of Rhombus Using Trigonometry
• Solved Problems on Area of Rhombus Formula
• Practice Questions on Area of Rhombus

What is the Area of a Rhombus?

The area of a rhombus is the total flat space enclosed within its four equal sides. In simple terms, if you draw a rhombus on paper and coloured it in, the amount of coloured surface would be its area. It is always measured in square units such as cm², m², or mm².

A rhombus looks like a diamond shape a four-sided figure where all sides are the same length, opposite sides run parallel to each other, and opposite angles are equal. The two diagonals cross each other at right angles inside, and that property is the key to finding its area.

Area of Rhombus Formula

Different formulas to find the area of a rhombus are tabulated below:

Where,

• d1 = length of diagonal 1
• d2 = length of diagonal 2
• b = length of any side
• h = height of rhombus
• a = measure of any interior angle

Derivation for Rhombus Area Formula

Consider the following rhombus: ABCD

Let O be the point of intersection of two diagonals AC and BD.

The area of the rhombus will be:

A = 4 × area of ∆ AOB

= 4 × (½) × AO × OB sq. units

= 4 × (½) × (½) d1 × (½) d2 sq. units

= 4 × (1/8) d1 × d2 square units

= ½ × d1 × d2

Therefore, the Area of a Rhombus = A = ½ × d1 × d2

Where d1 and d2 are the diagonals of the rhombus.

How to Calculate the Area of a Rhombus?

There are three methods to calculate the area of a rhombus are explained below with examples.

• Method 1: Using Diagonals
• Method 2: Using Base and Height
• Method 3: Using Trigonometry (i.e., using side and angle)

Method 1: Area of Rhombus Using Diagonals

Both diagonals are given.

Formula: Area = (d1 × d2) ÷ 2

Example: If d1 = 10 cm and d2 = 6 cm

Solution:

Area = (d1 × d2) ÷ 2

= (10 × 6) ÷ 2

= 60 ÷ 2 = 30 cm²

Method 2: Area of Rhombus Using Base and Height

The base (side length) and the perpendicular height are given.

Formula: Area = base × height

Example: If base = 8 cm and height = 5 cm

Solution:

Area = base × height

= 8 × 5 = 40 cm²

Method 3: Area of Rhombus Using Trigonometry

The side length and one interior angle are given, but the height is not directly known.

Formula: Area = side² × sin(angle)

Area = base × height = side × (side × sin(angle)) = side² × sin(angle)

Example: If side = 6 cm and angle = 30°

Solution:

Area = side² × sin(angle)

= 6² × sin(30°)

= 36 × 0.5 = 18 cm²

Solved Problems on Area of Rhombus Formula

1. Find Area Using Diagonals

The diagonals of a rhombus are 12 cm and 16 cm. Find its area.

Area = (d1 × d2) ÷ 2

= (12 × 16) ÷ 2

= 192 ÷ 2

= 96 cm²

2. Find Area Using Base and Height

A rhombus has a base of 9 cm and a perpendicular height of 7 cm. Find its area.

Area = base × height

= 9 × 7

= 63 cm²

3. Find Area Using Trigonometry

A rhombus has a side length of 10 cm and one interior angle of 60°. Find its area.

Area = side² × sin(angle)

= 10² × sin(60°)

= 100 × (√3/2)

= 100 × 0.866

= 86.6 cm²

4. Find a Diagonal When Area is Given

The area of a rhombus is 60 cm² and one diagonal is 10 cm. Find the other diagonal.

Area = (d1 × d2) ÷ 2

60 = (10 × d2) ÷ 2

60 × 2 = 10 × d2

120 = 10 × d2

d2 = 120 ÷ 10

= 12 cm

5. Find Side When Area and Angle are Given

The area of a rhombus is 50 cm² and one angle is 45°. Find the side length.

Area = side² × sin(angle)

50 = side² × sin(45°)

50 = side² × (1/√2)

50 = side² × 0.707

side² = 50 ÷ 0.707

side² = 70.72

side = √70.72

= 8.41 cm

6. Real-Life Problem

A diamond-shaped tile has diagonals of 8 cm and 6 cm. Find the area of the tile.

Area = (d1 × d2) ÷ 2

= (8 × 6) ÷ 2

= 48 ÷ 2

= 24 cm²

Practice Questions on Area of Rhombus

1. Find the area of a rhombus whose diagonals are 10 cm and 8 cm.
2. A rhombus has diagonals 12 cm and 16 cm. What is its area?
3. The area of a rhombus is 120 cm² and one diagonal is 15 cm. Find the other diagonal.
4. A rhombus has base 20 cm and area 200 cm². Find its height.
5. A rhombus has area 150 cm² and height 10 cm. Find the side length.