AB and CD are respectively arcs of two concentric circles of radii 21 cm and 7 cm and centre O.
Question:

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

Solution:

Radius of bigger circle $R=21 \mathrm{~cm}$ and sector angle $\theta=30^{\circ}$

$\therefore$ Area of the sector $\mathrm{OAB}$

$=\frac{30^{\circ}}{360^{\circ}} \times \frac{22}{7} \times 21 \times 21 \mathrm{~cm}^{2}$

$=\frac{11 \times 21}{2} \mathrm{~cm}^{2}=\frac{231}{2} \mathrm{~cm}^{2}$

Again, radius of the smaller circle, $r=7 \mathrm{~cm}$

Also, the sector angle is $30^{\circ}$

$\therefore$ Area of the sector OCD

$=\frac{\mathbf{3 0}^{\circ}}{\mathbf{3 6 0}^{\circ}} \times \frac{\mathbf{2 2}}{\mathbf{7}} \times 7 \times 7 \mathrm{~cm}^{2}=\frac{\mathbf{7 7}}{\mathbf{6}} \mathrm{cm}^{2}$

$\therefore$ Area of the shaded region $=$ Area of the sector OAB – Area of the sector OCD

$=\left[\frac{231}{2}-\frac{77}{6}\right] \mathrm{mn}^{2}=\frac{693-77}{6} \mathrm{~cm}^{2}$

$=\frac{616}{6} \operatorname{cm}^{2}=\frac{308}{3} \operatorname{cm}^{2}$

 

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