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$AB$ 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.
"
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 $\frac{308}{3} \mathrm{~cm}^{2}$.