Question:
In the following figure, the area of the shaded region is
(a) $3 \pi \mathrm{cm}^{2}$
(b) $6 \pi \mathrm{cm}^{2}$
(c) $9 \pi \mathrm{cm}^{2}$
(d) $7 \pi \mathrm{cm}^{2}$
Solution:
In the figure,
$\angle C=\angle B=90^{\circ}$ and $\angle D=60^{\circ}$,
$\therefore \angle A+\angle B+\angle C+\angle D=360^{\circ}$
$\angle A+90^{\circ}+90^{\circ}+60^{\circ}=360^{\circ}$
$\therefore \angle A=120^{\circ}$
Area of shaded region $=\frac{\theta}{360} \times \pi r^{2}$
$\therefore$ Area of shaded region $=\frac{120}{360} \times \pi \times 3^{2}$
$\therefore$ Area of shaded region $=\frac{1}{3} \times \pi \times 9$
$\therefore$ Area of shaded region $=3 \pi$
Therefore, area of the shaded region is $3 \pi \mathrm{cm}^{2}$
Hence, the correct answer is option (a).