In a circle of a radius 21 cm, an arc substends an angle of 60º at the centre. Find : (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord.

Here $$r \ = \ 21$$cm $$\theta \ = \ 60°$$
(i) Length of the arc, $$l \ = \ \frac{ \theta }{360} \ × \ 2\pi r$$

$$= \ \frac{60}{180} \ × \ \frac{22}{7}\ × \ 21 \ = \ 22$$cm

(ii) Area of the sector, $$A \ = \ \frac{ \theta }{360} \ × \ \pi r^2$$

$$= \ \frac{60}{360} \ × \ \frac{22}{7} \ × \ 21 \ × \ 21 \$$
$$= \ 231$$ cm2

(iii) Area of the segment

$$= \ r^2[ \frac{ \pi \theta }{360} \ - \ \frac{1}{2}sin \theta ]$$

$$= \ (21)^2[ \frac{22}{7} \ × \ \frac{60}{360} \ - \ \frac{1}{2}sin60°] \$$
$$= \ 441( \frac{11}{21} \ - \ \frac{1}{2} \ × \ \frac{ \sqrt{3}}{2})$$

$$= \ 21 \ × \ 11 \ - \ \frac{441 \sqrt{3}}{4} \$$
$$= \ (231 \ - \ \frac{441 \sqrt{3}}{4})$$cm2