The Area of a Rhombus

A rhombus is a quadrilateral having all sides with unequal diagonals which bisect each other.

Note: If a square is pressed from two opposite corners, a rhombus is formed.

Let ABCD be a rhombus, then its area can be evaluated in two ways.

 

(1) When one side and the included angle is given:

Let one side be equal to a with includes angle \theta . Since the diagonal AC divides the rhombus into two equal triangles, therefore


rhombus-01

The area of the rhombus  = 2 \times {\text{area of }}\Delta ABC.
The area of the rhombus  = 2\left( {\frac{1}{2}a \times a \times \sin \theta } \right) = 2\left( {\frac{1}{2}{a^2}} \right)\sin \theta = {a^2}\sin \theta .
The area of the rhombus  = {({\text{one side}})^2}\sin \theta .

 

(2) When two diagonal are given:

Let d and d' be the length of the diagonals AC and BD respectively, and since the rhombus is divided into four equal triangles, therefore


rhombus-02
 

The area of the rhombus  = 4 \times \frac{1}{2} \times \frac{{BD}}{2} \times \frac{{CA}}{2} = 4 \times \frac{{BD \times CA}}{8} = \frac{{BD \times CA}}{2}.

The area of the rhombus  = \frac{{AC \times BD}}{2} = \frac{{d \times d'}}{2}.

The area of the rhombus  = \frac{{{\text{Product of two diagonals}}}}{2}.

 

Example:

The length of each side of a rhombus is 120 cm and two of its opposite angles are 60^\circ each. Find the area.

 

Solution:

Given that each side a = 120 and \theta = 60^\circ ,
the area of the rhombus  = {({\text{side}})^2} \times \sin \theta .
The area of the rhombus  = 120 \times 120 \times \sin 60^\circ = 120 \times 120 \times 0.866 = 12470.4 square cm.

Example:

The diagonals of a rhombus are 40 m and 30 m. Find its area.

 

Solution:


rhombus-03

Given that the diagonals are d = 40 m and d' = 30 m,

the area of the rhombus  = \frac{{d \times d'}}{2} = \frac{{40 \times 30}}{2} = 600 square m.