# Area of 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, the rhombus is formed.

Let $ABCD$ be a rhombus then its area can be evaluated in two ways:

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

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

Area of the rhombus $= 2 \times {\text{area of }}\Delta ABC$
Area of the rhombus $= 2\left( {\frac{1}{2}a \times a \times \sin \theta } \right) = 2\left( {\frac{1}{2}{a^2}} \right)\sin \theta = {a^2}\sin \theta$
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 diagonal $AC$ and $BD$ respectively and since, the rhombus is divided into four equal triangles, therefore

Area of the rhombus $= 4 \times \frac{1}{2} \times \frac{{BD}}{2} \times \frac{{CA}}{2} = 4 \times \frac{{BD \times CA}}{8} = \frac{{BD \times CA}}{2}$

Area of the rhombus $= \frac{{AC \times BD}}{2} = \frac{{d \times d'}}{2}$

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 $\theta = 60^\circ$
Area of the rhombus $= {({\text{side}})^2} \times \sin \theta$
Area of the rhombus $= 120 \times 120 \times \sin 60^\circ = 120 \times 120 \times 0.866 = 12470.4$ square cm

Example:

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

Solution:

Given that diagonal are $d = 40$m and $d' = 30$m

Area of the rhombus $= \frac{{d \times d'}}{2} = \frac{{40 \times 30}}{2} = 600$ Square m