Question

Question 15
Area of sector of central angle ${200}^{\circ }$ of a circle is $770c{m}^2$. Find the length of the corresponding arc of this sector.

Answer 1

Let the radius of the sector AOBA be r.

Given: Central angle of sector AOBA $=\theta ={200}^{\circ }$

Area of the sector AOBA $=770c{m}^2$ [Given]

We know that, area of the sector $=\frac{\pi r^2}{{360}^{\circ }}\times {\theta }^{\circ }$

$\therefore$ Area of the sector, $770=\frac{\pi r^2}{{360}^{\circ }}\times {200}^{\circ }$ $\Rightarrow \frac{77\times 18}{\pi }=r^2$

$\Rightarrow r^2=\frac{77\times 18}{22}\times 7\Rightarrow r^2=9\times 49$

$\Rightarrow r=3\times 7$

$\therefore r=21cm$

So, radius of the sector AOBA = 21 cm.

Now, the length of the corresponding arc of this sector $=Centralangle\times Radius[\because \theta =\frac{l}{r}]$

$=200\times 21\times \frac{\pi }{{180}^{\circ }}[\because 1^{\circ }=\frac{\pi }{180}R]$

$=\frac{20}{18}\times 21\times \frac{22}{7}$

$=\frac{220}{3}cm=73\frac{1}{3}cm$

Hence, the required length of the corresponding arc is $73\frac{1}{3}cm$