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.


Answer :


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

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