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In figure, find the area of the shaded region, enclosed between two concentric circles of radii $7\ cm$ and $14\ cm$ where $\angle AOC\ =\ 40^{o}$.[Use $\pi =\frac{22}{7}$]
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Given: Two concentric circles of radii $7\ cm$ and $14\ cm$ where $\angle AOC=40^{o}$ in the given figure.

To do: To find the area of the shaded region, enclosed between two concentric circles

Solution:

Area of the region ABDC $=$Area of sector AOC $–$ Area of sector BOD

$=\frac{\theta }{360^{o}} \times \pi r^{2}_{1} -\frac{\theta }{360^{o}} \times \pi r^{2}_{2}$

$=\frac{40}{360^{o}} \times \frac{22}{7} \times ( 14)^{2} -\frac{40^{o}}{360^{o}} \times \frac{22}{7} \times ( 7)^{2}$

$=\frac{22}{9} \times ( 28-7)$

$=\frac{154}{3}$

$=51.33cm^{2}$

Area of circular ring $=$Area of the outer ring$-$Area of the inner ring

$=\pi r^{2}_{1} -\pi r^{2}_{2}$

$=\frac{22}{7}( 14^{2} -7^{2})$

$=22\ \times \ 14\ \times \ 2\ –\ 22\ \times \ 7\ \times \ 1$

$=22\ \times \ ( 28\ –\ 7)$

$=22\ \times \ 21$

$=462\ cm^{2}$

$\therefore$ Required shaded region $=$Area of circular ring$–$Area of region ABDC

$=462 – 51.33$

$=410.67\ cm^{2}$

Thus, The area of shaded region is $410.67\ cm^{2}$

 

Updated on: 10-Oct-2022

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