The area of a rhombus is the total space enclosed by the rhombus on a two-dimensional plane. The unit for the area is square units, such as sq. cm, sq. m, sq. ft, etc.

Formulas

There are 3 ways to find the area of a rhombus based on the parameter(s) given. The formulas are given below.

Using Diagonals

This is the most common method used to find the area of a rhombus. The formula to calculate the area of a rhombus when the diagonals are known is given below:

Derivation

Let us take a rhombus ABCD where O is the point of intersection of two diagonals d_{1} and d_{2}.

∆AOB is a right triangle as the diagonals of a rhombus are perpendicular bisectors.

The area of the rhombus will be 4 × area of ∆AOB. So the equation to derive the formula is written as:

= 4 × (½) × AO × OB

= 4 × (½) × (½) d_{1} × (½) d_{2}

= 4 × (1/8) d_{1} × d_{2}

= ½ × d_{1} × d_{2}

Let us solve an example to understand the concept better.

Calculate the area of a rhombus having diagonals equal to 9 inches and 8 inches.

Solution:

As we know, A = ½ × d_{1} × d_{2}, here d_{1} = 9 inches, d_{2} = 8 inches = ½ × 9 × 8 = 36 sq. inches

What is the area of rhombus ABCD.

Solution:

As we know, A = ½ × d_{1} × d_{2}, here d_{1} = 2 ×3 = 6 cm, d_{2} = 2 × 5 = 10 cm (∵ half of diagonals are 3 cm & 5 cm) = 6 × 10 = 60 sq. cm

Using Base and Height

The formula to calculate the area of a rhombus when the base and height are known is given below:

Let us solve an example to understand the concept better.

Finding the area of a rhombus when BASE and HEIGHT are known

What is the area of the rhombus with a base of 15 cm and a height of 11 cm?

Solution:

As we know, A = b × h, here b = 15 cm, h = 11 cm = 15 × 11 = 165 sq. cm

Using Side and Vertex Angle

The formula to calculate the area of a rhombus when side and any vertex angle are known is given below:

Let us solve an example to understand the concept better.

Finding the area of a rhombus when SIDE and VERTEX ANGLE are known

Find the area of a rhombus whose side is 18 cm and one of its angles is 30°.

Solution:

As we know, A = s^{2} × sin(θ), here s = 18 cm, θ = 30° = 18^{2} × sin(30) = 324 × ½ = 162 sq. cm