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=(1)/(12)*2\pi \\ =(\pi)/(6)\text{ units} \end{gathered}`

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

`\begin{gathered} A=(1)/(2)r^2\cdot\theta \\ =(1)/(2)r^2\cdot(l)/(r) \\ =(l\cdot r)/(2) \end{gathered}`

On plugging the values in the above relation,

`\begin{gathered} A=(\pi)/(6)*(1)/(2) \\ =(\pi)/(12) \\ =0.262\text{ sq. units} \end{gathered}`

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