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 forced by the corresponding chord

[use Π = 22/7]

Answer 1

Radius (r) of circle = 21 cm

Angle subtended by the given arc = 60°

Length of an arc of a sector of angle θ = `theta/360^@xx2pir`

Length of arc ACB = `(60^@)/360^@ xx2xx22/7xx21`

`= 1/6xx2xx22xx3`

= 22 cm

Area of sector OACB = `(60^@)/(360^@)xxpir^2`

`= 1/6xx22/7xx21xx21`

= 231 cm²

In ΔOAB,

∠OAB = ∠OBA (As OA = OB)

∠OAB + ∠AOB + ∠OBA = 180°

2∠OAB + 60° = 180°

∠OAB = 60°

Therefore, ΔOAB is an equilateral triangle.

Area of ΔOAB = `sqrt3/4 xx ("Side")^2`

`= sqrt3/4 xx (21)^2 = (441sqrt3)/4 cm^2`

Area of segment ACB = Area of sector OACB − Area of ΔOAB

`= (231 - (441sqrt3)/4) cm^2`