# $AB$ and $CD$ are respectively arcs of two concentric circles of radii 21 cm and 7 cm and centre O (see figure). If $\angle AOB=30^o$, find the area of the shaded region."

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Given:

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

$\angle AOB=30^o$.

To do:

We have to find the area of the shaded region.

Solution:

Radius of the sector $AOB = 21\ cm$

This implies,

Area of the sector $\mathrm{AOB}=\frac{\pi r^{2} \theta}{360^{\circ}}$

$=\frac{22}{7} \times \frac{21 \times 21 \times 30^{\circ}}{360^{\circ}}$

$=\frac{11 \times 21}{2}$

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

Radius of the sector $COD=7 \mathrm{~cm}$

$\angle \mathrm{COD}=30^{\circ}$

Therefore,

Area of the sector $COD=\frac{\pi r^{2} \theta}{360^{\circ}}$

$=\frac{22}{7} \times \frac{7 \times 7 \times 30^{\circ}}{360^{\circ}}$

$=\frac{77}{6} \mathrm{~cm}^{2}$

Area of the shaded region $=$ Area of the sector AOB $-$ Area of the sector COD

$=\frac{231}{2}-\frac{77}{6}$

$=\frac{693-77}{6}$

$=\frac{616}{6}$

$=\frac{308}{3} \mathrm{~cm}^{2}$

The area of the shaded region is $\frac{308}{3} \mathrm{~cm}^{2}$.

Updated on 10-Oct-2022 13:24:22