Question

A rhombus whose side measures 20 cm has one of the angles as ${60}^{\circ }.$. The area of the rhombus will be equal to

• A. $100\sqrt{3}c{m}^2$
• B. $200\sqrt{3}c{m}^2$
• C. $150\sqrt{3}c{m}^2$
• D. $250\sqrt{3}c{m}^2$

Answer 1

The correct option is B

$200\sqrt{3}c{m}^2$

Let the given rhombus be ABCD having $AB=AD=20cm$ and $\angle A={60}^{\circ }$

Draw DE perpendicular to AB

In $\Delta AED,\angle A={60}^{\circ },\angle AED={90}^{\circ },\angle ADE={30}^{\circ }$

The corresponding sides can be calculated as

$\Rightarrow \sin \left(30\right):\sin \left(60\right):\sin \left(90\right)$

$\Rightarrow \frac{1}{2}:\frac{\sqrt{3}}{2}:1$

$\Rightarrow 1:\sqrt{3}:2$

$\begin{array}{ccccc}{30}^{\circ }& & {60}^{\circ }& & {90}^{\circ }\\ x& :& x\sqrt{3}& :& 2x\\ \downarrow & & \downarrow & & \downarrow \\ AE& & DE& & AD\\ \downarrow & & \downarrow & & \downarrow \\ 10cm& & 10\sqrt{3}cm& & 20cm\end{array}$

Area of rhombus = $AB\times DE$

$=20\times 10\sqrt{3}$

$=200\sqrt{3}c{m}^2$