What will be the length of the arc of a circle of radius 21 cm which substance and angle of 60 degree at the centre?

Given

A circle of radius 21 cm, an arc subtends an angle of 60° at the centre.

Find out

(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

Solution:

(i) Radius = r = 21 cm Central angle = θ = 60° Length of the arc APB = (θ/360°) × 2πr = (60°/360°) × 2 × (22/7) × 21 = (1/6) × 2 × 22 × 3 = 22

Therefore, the length of the arc = 22 cm

(ii) Area of sector OAPB = (θ/360°) × πr2 = (60°/360°) × (22/7) × 21 × 21 = (1/6) × 22 × 3 × 21 = 11 × 21 = 231

Therefore, the area of the sector = 231 cm2

(iii) Area of the segment APB = Area of sector OAPB – Area of triangle OAB
= 231 – (√3/4) (side)2 {since OA = OB (radii) and the angles corresponding to these sides are equal, thus the triangle OAB becomes an equilateral triangle} = 231 – (√3/4) × 21 × 21

= 231 – (441√3)/4

= 231 – (441 × 1.73)/4

= 231 – 190.7325

= 40.2675

Therefore, the area of the corresponding segment = 40.2675 cm2

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