Question

In a circle of radius 21 cm, an arc subtends an angle of 60° 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 1

Radius of the circle (r) = 21 cm
Angle subtended by the arc = 60°

(i) Length of an arc of a sector with angle θ = \(\frac {\theta }{360^{\degree}} \times 2 \pi r\)

Length of arc ACB = \(\frac{60°}{360 °} \times 2 \times \frac{22}7 \times 21\)
= \(\frac{1}{6} \times 2 \times {22} \times 3\)
= 22 cm

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(ii) Area of sector OACB = \(\frac{60°}{360 °} \times \pi r^2\)

= \(\frac{1}{6} \times \frac{22}7 \times 21\times 21\)
= \(231 cm ^2\)

In triangle OAB,

∠OAB = ∠OBA (Since OA = OB)
∠OAB + ∠AOB + ∠OBA = 180°
2∠OAB + 60° = 180°
∠OAB = 60°
Therefore, triangle OAB is an equilateral triangle.

Area of triangle OAB = \(\frac{ \sqrt3 }{4} \times (Side) ^2\)

= \(\frac{ \sqrt3 }{4} \times (21) ^2 = \frac{441 \sqrt 3}{4} \, cm^2\)

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(iii) Area of segment ACB = Area of sector OACB - Area of triangle OAB
= \((231 - \frac{441 \sqrt3}{4})\, cm^2\)