Question

In Fig. 7, PQ and AB are respectively the arcs of two concentric circles of radii 7 cm and 3.5 cm and centre O. If $\angle POQ={30}^{\circ }$ , then find the area of the shaded region. [Use $=\frac{22}{7}$]

Answer 1

PQ and AB are the arcs of two concentric circles of radii 7 cm and 3.5 cm respectively.
Let $r_1$ and $r_2$ be the radii of the outer and the inner circle respectively.
Suppose $\theta$ be the angle subtended by the arcs at the centre O.
THen $r_1=7cm,r_2=3.5cm$ and $\theta ={30}^{\circ }$
Area of the shaded region
=Area of sector OPQ -Area of sector OAB
$=\frac{\theta }{{360}^{\circ }}\pi r_1^2-\frac{\theta }{{360}^{\circ }}\pi r_2^2=\frac{\theta }{{360}^{\circ }}\pi (r_1^2-r_2^2)=\frac{{30}^{\circ }}{{360}^{\circ }}\times \frac{22}{7}\left[\right(7cm{)}^2-\left(3.5cm{)}^2\right]=\frac{1}{12}\times \frac{22}{7}\times (49-12.25)c{m}^2=\frac{1}{12}\times \frac{22}{7}\times 36.75c{m}^2=9.625c{m}^2$
thus, the area of the shaded region is $9.625c{m}^2$.