Question

In a circle of radius 14 cm, an arc subtends an angle of 120° at the centre. If $\sqrt{3}$= 1.73 then the area of the segment of the circle is
(a) 120.56 cm²
(b) 124.63 cm²
(c) 118.24 cm²
(d) 130.57 cm²

Answer 1

Radius of the circle, r = 14 cm
Draw a perpendicular OD to chord AB. It will bisect AB.
∠A = 180° − (90° + 60°) = 30°

$$\begin{gathered} \cos 30°=\frac{\mathrm{AD}}{\mathrm{OA}} \\ \Rightarrow \frac{\sqrt{3}}{2}=\frac{\mathrm{AD}}{14} \\ \Rightarrow \mathrm{AD}=7\sqrt{3} \\ \Rightarrow \mathrm{AB}=2\times \mathrm{AD}=14\sqrt{3}\mathrm{cm} \\ \sin 30°=\frac{\mathrm{OD}}{\mathrm{OA}} \\ \Rightarrow \frac{1}{2}=\frac{\mathrm{OD}}{14} \\ \Rightarrow \mathrm{OD}=7\mathrm{cm} \end{gathered}$$

Area of minor segment = Area of sector OAPB − Area of triangle AOB
$$\begin{gathered} =\frac{\mathrm{\theta }}{360°}\mathrm{\pi }{\left(\mathrm{OA}\right)}^2-\frac{1}{2}\times \mathrm{OD}\times \mathrm{AB} \\ =\frac{120°}{360°}\times \frac{22}{7}{\left(14\right)}^2-\frac{1}{2}\times 7\times 14\sqrt{3} \\ =205.33-84.77 \\ =120.56{\mathrm{cm}}^2 \end{gathered}$$
Hence, the correct answer is option (a).