Question

A chord PQ of length 12 cm subtends an angle of ${120}^{\circ }$ at the centre of a circle. Find the area of the minor segment cut off by the chord PQ.

Answer 1

Area of the shaded segment $=\pi r^2\times \frac{\theta }{{360}^o}-r^{2}sin\frac{\theta }{2}cos\frac{\theta }{2}$

$=\pi \times 12\times 12\times \frac{120}{360}-12\times 12\times sin{60}^{o}cos{60}^o$

$=48\pi -144\times \frac{\sqrt{3}}{2}\times \frac{1}{2}$

$=48\pi -36\sqrt{3}$

$=12(4\pi -3\sqrt{3})$