Question

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

Answer 1

In the mentioned figure,

$O$ is the centre of circle,

$AB$ is a chord

$AXB$ is a major arc,

$OA=OB=$ radius $=21$ cm

Arc $AXB$ subtends an angle ${60}^o$ at $O.$

i) Length of an arc $AXB$ $=\frac{60}{360}\times 2\pi \times r$

$=\frac{1}{6}\times 2\times \frac{22}{7}\times 21$

$=22cm$

ii) Area of sector $AOB$ $=\frac{60}{360}\times \pi \times r^2$

$=\frac{1}{6}\times \frac{22}{7}\times (21{)}^2$

$=231c{m}^2$

iii) Area of segment (Area of Shaded region) $=$ Area of sector $AOB-$ Area of $△AOB$

By trigonometry,

$AC=21\sin 30$

$OC=21\cos 30$

And, $AB=2AC$

$\therefore$ $AB=42\sin 30=41\times \frac{1}{2}=21$ cm

$\therefore$ $OC=21\cos 30=\frac{21\sqrt{3}}{2}$ cm

$\therefore$ Area of $△$ AOB $=\frac{1}{2}\times AB\times OC$

$=\frac{1}{2}\times 21\times \frac{21\sqrt{3}}{2}=\frac{441\sqrt{3}}{4}{cm}^2$

$\therefore$ Area of segment (Area of Shaded region) $=(231-\frac{441\sqrt{3}}{4}){cm}^2$