Question

In the unit circle, if the arc length is 1/20 of the circumference, find the area of the sector.

Answer 1

Given:

The length of the arc is (1/12) x circumference of unit circle.

The objective is to find the area of the sector.

Since it is given as a unit cirle, the radius of the circle will be 1 unit.

The circumference of the circle will be,

$\begin{gathered} C=2\cdot\pi\cdot r \\ =2\pi\text{ units.} \end{gathered}$

Then, the length of the arc will be,

$\begin{gathered} l=\frac{1}{12}\times2\pi \\ =\frac{\pi}{6}\text{ units} \end{gathered}$

Now, the formula to find the area of the sector is,

$\begin{gathered} A=\frac{1}{2}r^2\cdot\theta \\ =\frac{1}{2}r^2\cdot\frac{l}{r} \\ =\frac{l\cdot r}{2} \end{gathered}$

On plugging the values in the above relation,

$\begin{gathered} A=\frac{\pi}{6}\times\frac{1}{2} \\ =\frac{\pi}{12} \\ =0.262\text{ sq. units} \end{gathered}$

Hence, the area of the sector is 0.262 square units.