In a circle of radius 21 cm, an arc subtends an angle of 60° at the center. 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

We know that,  formula for the area of the sector of a circle

Area of the sector = $\frac{\theta }{{360}^0}$ $\times$ ${\mathrm{\pi r}}^2$

 Area of the segment  = Area of the sector - Area of the corresponding  triangle

Length of the arc = $\frac{\theta }{{360}^0}$ $\times$ 2$\mathrm{\pi }$r

Here, r = 21 cm, θ = 60°

Area of the segment APB = Area of sector AOPB - Area of ΔAOB

(i) Length of the arc APB = 

Length of the arc APB  = $\frac{\theta }{{360}^0}$ $\times$ 2$\mathrm{\pi }$r

= 60°/360° $\times$ 2 $\times$ 22/7 $\times$ 21

= 22 cm

(ii)Area of the sector = $\frac{\theta }{{360}^0}$ $\times$ ${\mathrm{\pi r}}^2$

Area of the sector AOBP= $\frac{\theta }{{360}^0}$ $\times$ ${\mathrm{\pi r}}^2$

= 60°/360° $\times$ 22/7 $\times$ 21 $\times$ 21

= 231 cm²

 (iii) Area of the segment  = Area of the sector - Area of the corresponding  triangle:

Area of the corresponding  triangle AOB = 1/2 $\times$ AB $\times$OM = 1/2 $\times$ r $\times$$\frac{\sqrt{3}}{2}$

= 441 $\frac{\sqrt{3}}{4}$ cm²

Area of the segment = (231 -  441 $\frac{\sqrt{3}}{4}$ )cm²