In the figure below, two circles with centres $ A $ and $ B $ touch each other at the point $ C $. If $ A C=8 \mathrm{~cm} $ and $ A B=3 \mathrm{~cm} $, find the area of the shaded region. "
Given:
Two circles with centres \( A \) and \( B \) touch each other at the point \( C \).
\( A C=8 \mathrm{~cm} \) and \( A B=3 \mathrm{~cm} \).
To do:
We have to find the area of the shaded region.
Solution:
$BC = 8 - 3\ cm$
$= 5\ cm$
Radius of the bigger circle $R= 8\ cm$
Radius of the smaller circle $r = 5\ cm$
Therefore,
Area of the shaded region $=$ Area of the bigger circle $-$ Area of the smaller circle
$=\pi R^{2}-\pi r^{2}$
$=\frac{22}{7}(8^{2}-5^{2})$
$=\frac{22}{7}(64-25)$
$=\frac{22}{7} (39)$
$=122.57 \mathrm{~cm}^{2}$
The area of the shaded region is $122.57\ cm^2$.
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