Question

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

Answer 1

Radii of the concentric circles = 21 cm and 42 cm
Area between the circles $=\pi (R^2-r^2)=\left(\frac{22}{7}\right)({42}^2-{21}^2)=4158c{m}^2$

Angle subtended by the arc in the inner circle $={60}^o$
Area of the sector in the inner circle $=\left(\frac{{60}^o}{{360}^o}\right)\times \pi r^2=\left(\frac{{60}^o}{{360}^o}\right)\times \left(\frac{22}{7}\right)\times (21{)}^2=231c{m}^2$

Angle subtended by the arc in the outer circle $={60}^o$
Area of the sector in the inner circle $=\left(\frac{{60}^o}{{360}^o}\right)\times \pi R^2=\left(\frac{{60}^o}{{360}^o}\right)\times \left(\frac{22}{7}\right)\times (42{)}^2=924c{m}^2$

Area of the portion of the sector in between the circles $=924-231=693c{m}^2$

Area of the shaded portion = (Area between the circles) - (Area of the portion of the sector in between the circles)
$=4158-693=3465$

Therefore, the area of the shaded region is 3465 $c{m}^2$