Question

In the given figure, O is the centre of the bigger circle, and AC is its diameter. Another circle with AB as diameter is drawn. If AC = 54 cm and BC = 10, find the area of the shaded region.

Answer 1

We have:
OA = OC = 27 cm
AB = AC $-$ BC
= 54 $-$ 10
= 44
AB is the diameter of the smaller circle.
Thus, we have:
Radius of the smaller circle = $\frac{\mathrm{AB}}{2}=\frac{44}{2}=22\mathrm{cm}$
Area of the smaller circle = ${\mathrm{\pi r}}^2$
$$\begin{gathered} =\frac{22}{7}\times 22\times 22 \\ =1521.14{\mathrm{cm}}^2 \end{gathered}$$

Radius of the larger circle = $\frac{\mathrm{AC}}{2}=\frac{54}{2}=27\mathrm{cm}$
Area of the larger circle = ${\mathrm{\pi r}}^2$
$$\begin{gathered} =\frac{22}{7}\times 27\times 27 \\ =2291.14{\mathrm{cm}}^2 \end{gathered}$$
∴ Area of the shaded region = Area of the larger circle $-$ Area of the smaller circle
= 2291.14 $-$ 1521.14
= 770 cm²