Question

OPQ is the sector of a circle having
centre at 0 and radius 15 cm. If
mZPOQ = 30°
30°, find the area enclosed
by arc PQ and chord PQ.​

Answer 1

Answer: $2.68\,\rm{cm}^2$

Step-by-step explanation:

Given:

The radius is, $r=15\,\rm{cm}$.

The angle made by the arc at the center of the circle is $\theta=30^\circ$.

Calculate the area of the sector.

$\begin{aligned}A&=\pi r^2\frac{\theta}{360^\circ}\\&=\frac{22}{7} (15)^2\frac{30^\circ}{360^\circ}\\&=\frac{22}{7}(225)\frac{1}{12}\\&=\frac{825}{14} \end{aligned}$

Calculate the area of triangle POQ.

$\begin{aligned}A_1&=\frac{1}{2}r^2\sin\theta\\&=\frac{1}{2}(15)^2\sin30^{\circ}\\&=\frac{225}{4}\end{aligned}$

Calculate the area enclosed between the arc PQ and the chord PQ.

$\begin{aligned}A_2&=A-A_1\\&=\frac{825}{14}-\frac{225}{4}\\&=\frac{1650-1575}{28}\\&=\frac{75}{28}\\&=2.68\,\rm{cm}^2\end{aligned}$