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
Radius = 21 cm
θ = 60°
(i) Length of an arc = θ/360° × Circumference (2πr)
∴ Length of an arc AB = (60°/360°) × 2 × (22/7) × 21
= (1/6) × 2 × (22/7) × 21
Arc AB Length = 22cm
(ii) It is given that the angle subtend by the arc = 60°
Area of the sector making an angle of 60° = (60°/360°) × π r2 cm2
= 441/6 × 22/7 cm2
The area of the sector formed by the arc APB is 231 cm2
(iii) Area of segment APB = Area of sector OAPB – Area of ΔOAB
Since the two arms of the triangle are the radii of the circle and thus are equal, and one angle is 60°, ΔOAB is an equilateral triangle. So, its area will be √3/4 × a2 sq. Units.
Area of segment APB = 231 - (√3/4) × (OA)2
= 231 - (√3/4) × 212
Area of segment APB = [231 - (441 × √3)/4] cm2