Question

Find the area of the shaded region in figure, if radii of the two concentric circles with center $O$ are 7 cm and 14 cm respectively and $\angle AOC={40}^{\circ }$.

Answer 1

Given that,

Radius of the small circle, $OB=7cm$

Radius of second circle, $OA=14cm$

and $\angle AOC={40}^o$

We know that, the area of a sector that subtends an angle $\theta$ at the centre of the circle is $\frac{\theta }{{360}^0}\times \pi r^2$

where, $\theta$ is in degrees.

$\therefore$ Area of minor sector $OBD=\frac{40}{360}\times \frac{22}{7}\times 7\times 7[\because \pi =\frac{22}{7}]$

$=\frac{1}{9}\times 22\times 7$

$=17.11c{m}^2$

Also, area of minor sector $OAC=\frac{40}{360}\times \frac{22}{7}\times 14\times 14$

$=\frac{1}{9}\times 22\times 2\times 14$

$=68.4c{m}^2$

Now, area of the shaded region $=$ Area of sector $OAC-$ Area of sector $OBD$

$=68.4-17.1$

$=51.3c{m}^2$

Hence, the area of the shaded region is $51.3c{m}^2$.