Question

In the given figure, PQ and AB are respectively the arcs of two concentric circles of radii 7 cm and 3.5 cm with centre O. If ∠POQ = 30°, find the area of the shaded region. [CBSE 2012]

Answer 1

Area of the shaded portion = Area of sector OPQ − Area of sector OAB
$$\begin{gathered} =\frac{30°}{{360}^{°}}\times \mathrm{\pi }{\left(7\right)}^2-\frac{30°}{{360}^{°}}\times \mathrm{\pi }{\left(3.5\right)}^2 \\ =\frac{22}{7}\times \frac{1}{12}\left[{\left(7\right)}^2-{\left(3.5\right)}^2\right] \\ =\frac{22}{7}\times \frac{1}{12}\left[{\left(7\right)}^2-{\left(\frac{7}{2}\right)}^2\right] \\ =\frac{22}{7}\times \frac{1}{12}\times \frac{147}{4} \\ =\frac{77}{8}{\mathrm{cm}}^2 \end{gathered}$$
Hence, the area of the shaded portion is $\frac{77}{8}$ cm².