Question

The area of $\odot (O,r)$ is $240c{m}^2$. If the minor $\widehat{ACB}$ subtends an angle of ${45}^{\circ }$ then the area of the minor sector $OACB$ is _____ $c{m}^2$.

• A. $80$
• B. $40$
• C. $60$
• D. $30$

Answer 1

The correct option is D $30$

The area of $\odot (O,r)=240c{m}^2$
$\therefore \pi r^2=240$
$\therefore r^2=\frac{240}{\pi }....\left(1\right)$
The minor $\widehat{ACB}$ subtends an angle of ${45}^{\circ }$ at the centre of the circle.
$\theta ={45}^{\circ }.....\left(2\right)$
The area of the minor sector $OACD$
$=\frac{\pi r^2\theta }{360}$
$=\frac{\pi }{360}\times \frac{240}{\pi }=45$
(from result $\left(1\right)$ and $\left(2\right)$)
$=30c{m}^2$.