Find the area of the shaded region in the given figure, if radii of the two concentric circles with centre $O$ are $7\ cm$ and $14\ cm$ respectively and $\angle AOC = 40^o$
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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:

We have 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

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

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

$=\frac{22}{7} \times \frac{1}{9}\times( 196-49)$

$=\frac{22\times147}{7\times9}$

$=\frac{154}{3}$

$=51.33cm^{2}$

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

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

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

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

$=22 \times (21)$

$=22\ \times \ 21$

$=462\ cm^{2}$

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

$=462 – 51.33$

$=410.67\ cm^{2}$

Therefore, the area of shaded region is $410.67\ cm^{2}$.

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Updated on: 10-Oct-2022

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