Question

If the area of a sector is one fourth of the area of the circle, then the angle of the sector is:

• A. ${180}^{\circ }$
• B. ${40}^{\circ }$
• C. ${45}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is D ${90}^{\circ }$

$\text{Area of the circle =}\pi r^2$
$\text{Area of the sector =}\frac{\theta }{{360}^{\circ }}\times \pi r^2,$ where $\theta$ is the angle of the sector

Given, the area of a sector is one fourth of the area of the circle.
$\therefore \frac{\theta }{{360}^{\circ }}\times \pi r^2=\frac{1}{4}\times \pi r^2$
Cancel out $\pi r^2$ from both sides
$\Rightarrow \frac{\theta }{{360}^{\circ }}=\frac{1}{4}$
Multiply both sides by ${360}^{\circ }$
$\Rightarrow \frac{\theta }{{360}^{\circ }}\times {360}^{\circ }=\frac{1}{4}\times {360}^{\circ }$
$\Rightarrow \theta =\frac{1}{4}\times 4\times {90}^{\circ }$
$\Rightarrow \theta ={90}^{\circ }$

Option (d.) is the correct choice.