Question

In the given figure, two concentric circles with centre $O$ have radii $21$ cm and $42$ cm, If $\angle AOB={60}^o$, find the area of the shaded region.
$[\text{Use}\pi =\frac{22}{7}]$

Answer 1

Given: Radius of inner circle $=r=21$ cm

Radius of outer circle $=R=42$ cm

Angle made by sector = ${60}^o$

Area of sector OAB = $\frac{{60}^o}{{360}^o}\times \pi R^2$

$=\frac{1}{6}\times \frac{22}{7}\times {42}^2=924c{m}^2$

Area of sector OCD = $\frac{{60}^o}{{360}^o}\times \pi r^2$

$=\frac{1}{6}\times \frac{22}{7}\times {21}^2=231c{m}^2$

Area of the small region $ABDC$

$=$ Area of the small sector $OAB$ $–$ Area of the small sector $OCD$ $=(923.628-230.907)c{m}^2$

Area of shaded region $ABDC=$ Area of outer circle $-$ Area of inner circle $-$ area of small region $ABDC$

Area of shaded region $ABDC=\pi \times {42}^2-\pi \times {21}^2-(924-231)$ cm${}^2$

Area of shaded region $ABDC=693$ cm${}^2$