Question

In a circle of radius $21$cm, an arc subtends an angle of ${60}^o$ 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

Answer 1

(i) Length of the arc $=\frac{\theta }{360}\times 2\pi r$

$=\frac{60}{360}\times 2\times \frac{22}{7}\times 21$

$=22cm$ [Ref. image (i)]

(ii) Area $=\frac{\theta }{360}\pi r^2$

$=\frac{60}{160}\times \frac{22}{7}\times 21\times 21$

$=231c{m}^2$

(iii) Area of sector $APB=\Delta OAPB-\Delta OAB$

find $\Delta$ of $AOB$ [Ref. image (iii)]

$OM=\pi \times \cos 30$

$=21\times \frac{\sqrt{3}}{2}$

Area of $\Delta AOB=\frac{1}{2}\times Base\times height$

$=\frac{1}{2}\times 21\times \frac{\sqrt{3}}{2}\times 21$

$=\frac{441\sqrt{3}}{3}$

Area of segment $=(231-\frac{441\sqrt{3}}{4})c{m}^2$