Question

How to find the value of k?

AB is an arc of a circle , centre O , radius 9 cm.
The length of the arc AB is `6picm`.
The area of the sector AOB is `kpicm^2` .
Find the value of k.

Answer 1

`k=27`

Explanation:

The circumference of a circle of radius `r`, which is an arc subtending an angle of `2pi`, is `2pir`.

So if the angle subtended by an arc is `theta` then the length of the arc is `rtheta`, where `r` is the radius.

The area of a circle of radius `r`, which is a sector subtending an angle of `2pi`, is `pir^2 = 1/2(2pi)r^2`.

So if the angle subtended by a sector is `theta` then the area of the sector is `1/2thetar^2`, where `r` is the radius.

In our example, we have `r=9cm` and length of arc `6picm`.

So (in `cm`) we have:

`9theta = 6pi`

Hence:

`theta = (6pi)/9 = (2pi)/3`

Then the area (in `cm^2`) of the sector is:

`kpi = 1/2thetar^2 = 1/2((2pi)/3)(9^2) = 27pi`

So:

`k=27`