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:
Area of Rhombus
Derivation
Let us take a rhombus ABCD where O is the point of intersection of two diagonals d1 and d2.
Derivation of Area of Rhombus Formula
∆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) d1 × (½) d2
= 4 × (1/8) d1 × d2
= ½ × d1 × d2
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 × d1 × d2, here d1 = 9 inches, d2 = 8 inches = ½ × 9 × 8 = 36 sq. inches
What is the area of rhombus ABCD.
Solution:
As we know, A = ½ × d1 × d2, here d1 = 2 ×3 = 6 cm, d2 = 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:
Area of Rhombus Using Base and Height
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:
Area of Rhombus Using Side and Angle
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 = s2 × sin(θ), here s = 18 cm, θ = 30° = 182 × sin(30) = 324 × ½ = 162 sq. cm
What is the area of rhombus ABCD.
