Question

$AB$ and $CD$ are respectively arcs of two concentric circles of radii $21cm$ and $7cm$ and centre $O$. If $\angle AOB={30}^{\circ }$, find the area of the shaded region.

Answer 1

Given:

Radius of the larger sector, $R=21cm$
Radius of the smaller sector, $r=7cm$
Angle subtended by sectors of both concentric circles $={30}^{\circ }$

Area of the sector making angle $\theta$
$=\left(\frac{\theta }{{360}^{\circ }}\right)\times \pi r^2$

Area of the larger sector
$=\left(\frac{{30}^{\circ }}{{360}^{\circ }}\right)\times \pi R^{2}c{m}^2$

$=\frac{1}{12}\times \frac{22}{7}\times {21}^{2}c{m}^2$

$=\frac{1}{12}\times \frac{22}{7}\times 21\times 21c{m}^2$

$=\frac{231}{2}c{m}^2$

Area of the smaller circle
$=\left(\frac{{30}^{\circ }}{{360}^{\circ }}\right)\times \pi r^{2}c{m}^2$

$=\frac{1}{12}\times \frac{22}{7}\times 7^{2}c{m}^2$

$=\frac{1}{12}\times \frac{22}{7}\times 7\times 7c{m}^2$

$=\frac{77}{6}c{m}^2$

Area of the shaded region $=$ Area of the larger sector $-$ Area of the smaller sector
$=(\frac{231}{2}-\frac{77}{6})c{m}^2$

$=\frac{616}{6}c{m}^2$

$=\frac{308}{3}c{m}^2$

$=102\frac{2}{3}c{m}^2$