$AB$ and $CD$ are respectively arcs of two concentric circles of radii $21\, cm$ and $7\, cm$ with centre $O$ (see figure). If $\angle AOB = 30^{\circ}$,find the area of the shaded region.
(N/A) The area of the shaded region is the difference between the area of the larger sector $OAB$ and the smaller sector $OCD$.
Area of sector $OAB = \frac{\theta}{360^{\circ}} \times \pi R^2 = \frac{30^{\circ}}{360^{\circ}} \times \frac{22}{7} \times (21)^2 = \frac{1}{12} \times \frac{22}{7} \times 441 = \frac{1}{12} \times 22 \times 63 = \frac{1386}{12} = 115.5\, cm^2$.
Area of sector $OCD = \frac{\theta}{360^{\circ}} \times \pi r^2 = \frac{30^{\circ}}{360^{\circ}} \times \frac{22}{7} \times (7)^2 = \frac{1}{12} \times \frac{22}{7} \times 49 = \frac{1}{12} \times 22 \times 7 = \frac{154}{12} = \frac{77}{6}\, cm^2$.
Area of the shaded region = Area of sector $OAB$ $-$ Area of sector $OCD = 115.5 - \frac{77}{6} = \frac{693 - 77}{6} = \frac{616}{6} = \frac{308}{3}\, cm^2$.
Area of sector $OAB = \frac{\theta}{360^{\circ}} \times \pi R^2 = \frac{30^{\circ}}{360^{\circ}} \times \frac{22}{7} \times (21)^2 = \frac{1}{12} \times \frac{22}{7} \times 441 = \frac{1}{12} \times 22 \times 63 = \frac{1386}{12} = 115.5\, cm^2$.
Area of sector $OCD = \frac{\theta}{360^{\circ}} \times \pi r^2 = \frac{30^{\circ}}{360^{\circ}} \times \frac{22}{7} \times (7)^2 = \frac{1}{12} \times \frac{22}{7} \times 49 = \frac{1}{12} \times 22 \times 7 = \frac{154}{12} = \frac{77}{6}\, cm^2$.
Area of the shaded region = Area of sector $OAB$ $-$ Area of sector $OCD = 115.5 - \frac{77}{6} = \frac{693 - 77}{6} = \frac{616}{6} = \frac{308}{3}\, cm^2$.
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