Question

In the figure, $PQ$ and $AB$ are respectively the arcs of two concentric circles of radii $7$ cm and $3.5$ cm and centre O. If $\angle POQ={30}^0$, then the area of the shaded region. $[Use\pi =\frac{22}{7}].$

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 $r_1=7$ cm, $r_2=3.5$ cm and $\theta ={30}^o$

Area of the shaded region $=$ Area of sector OPQ $-$ Area of sector $OAB$

$=\frac{\theta }{{360}^o}\pi r_1^2-\frac{\theta }{{360}^o}\pi r_2^2$

$=\frac{\theta }{{360}^o}\pi (r_1^2-r_2^2)$

$=\frac{{30}^o}{{360}^o}\times \frac{22}{7}[7^2-{3.5}^2]$

$=\frac{1}{12}\times \frac{22}{7}\times (49-12.25)$

$=\frac{1}{12}\times \frac{22}{7}\times 36.75$

$=9.625$

Therefore, the area of the shaded region is $9.625$ $c{m}^2$.