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 $122.57\ cm^2$.