Area of a sector of central angle \( 200^{\circ} \) of a circle is \( 770 \mathrm{~cm}^{2} \). Find the length of the corresponding arc of this sector.

Given:

Area of a sector of central angle \( 200^{\circ} \) of a circle is \( 770 \mathrm{~cm}^{2} \).

To do:

We have to find the length of the corresponding arc of this sector.

Solution:

Let the radius of the circle be $r$ and the central angle of the sector be $\theta$.

This implies,

$\theta= 200^o$

Area of the sector $= 770 cm^2$

We know that,

Area of a sector $=\frac{\pi r^{2}}{360^{\circ}} \times \theta^{\circ}$

Therefore,

$\frac{\pi r^{2}}{360^{\circ}} \times 200=770$
$\Rightarrow \frac{77 \times 18}{\pi}=r^{2}$

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

$\Rightarrow r^{2}=9 \times 49$

$\Rightarrow r=3 \times 7$

$\Rightarrow r=21 \mathrm{~cm}$

The radius of the sector is $21 \mathrm{~cm}$

The length of the corresponding arc of the sector $=$ Central angle $\times$ Radius

$=200 \times 21 \times \frac{\pi}{180^{\circ}}$

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

$=\frac{220}{3} \mathrm{~cm}$

$=73 \frac{1}{3} \mathrm{~cm}$

The required length of the corresponding arc is $73 \frac{1}{3} \mathrm{~cm}$.

Updated on: 10-Oct-2022

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