In the given figure, two concentric circles with center $ O $ have radii $ 21\;{\text{cm}} $ and $ 42\;{\text{cm}} $ , If $ \angle AOB = 60^\circ $ , find the area of the shaded region.

Hint: Use the formula for the area of the sector and find the area of the major sector of the bigger circle and then the smaller circle and then subtract both the areas to get the required area.

Complete step-by-step answer:

As the angle of the minor sector for both the bigger and smaller circles is equal to $ 60^\circ $ . So, the angle of the major sector for both the bigger and smaller circles is equal to $ 300^\circ $ .
The formula for the area of the sector of circle is equal to $ \dfrac{\theta }{{360}}\pi {r^2} $ , where $ \theta $ is the angle of the sector and $ r $ is the radius of the circle.
The radius of the bigger circle is equal to \[42\;{\text{cm}}\] and the angle for the major sector of this circle is equal to $ 300^\circ $ .
The area of the major sector of bigger circle is:
 $
  A = \dfrac{\theta }{{360}}\pi {r^2} \\
   = \dfrac{{300}}{{360}} \times 3.14 \times {\left( {42\;{\text{cm}}} \right)^2} \\
   = \dfrac{5}{6} \times 3.14 \times 1764 \\
   = 4620\;{\text{c}}{{\text{m}}^2} \\
  $
The radius of the smaller circle is equal to \[21\;{\text{cm}}\] and the angle for the major sector of this circle is equal to $ 300^\circ $ .
The area of the major sector of smaller circle is:
 $
  A = \dfrac{\theta }{{360}}\pi {r^2} \\
   = \dfrac{{300}}{{360}} \times 3.14 \times {\left( {21\;{\text{cm}}} \right)^2} \\
   = \dfrac{5}{6} \times 3.14 \times 441 \\
   = 1155\;{\text{c}}{{\text{m}}^2} \\
  $
The area of the shaded region is the difference of the area of the major sector of the bigger circle and the area of the major sector of the smaller circle.
So, the area of the shaded region is equal to $ 4620\;{\text{c}}{{\text{m}}^2} - 1155\;{\text{c}}{{\text{m}}^2} = 3465\;{\text{c}}{{\text{m}}^2} $ .
Therefore, the area of the shaded region is equal to $ 3465\;{\text{c}}{{\text{m}}^2} $ .
So, the correct answer is “ $ 3465\;{\text{c}}{{\text{m}}^2} $ ”.

Note: Use the formula for the area of any sector of circle with angle $ \theta $ and radius $ r $ is equal to $ \dfrac{\theta }{{360}}\pi {r^2} $ . As observed from the diagram the shaded region is the area of the major sector of the bigger circle except the area of the major sector of the smaller circle.