2. Important formulae

In this section, let us look at a few important formulae related to sector and segment of a circle.

 Let us first look at the image given below for a thorough understanding.

 

 

Here, $OAPB$ is the sector of a circle with '$r$' as the radius. Also, let $\angle AOB$ be $\theta$.

 

We are well aware that the area of a circle is $\pi r^2$.

 

We also know that any circular region, with $O$ as the centre, is a sector forming the angle $360^\circ$.

 

In other words, the area of a sector forming $360^\circ$ is $\pi r^2$.

 

So, the area of a sector forming $1^\circ = \frac{1}{360} \times \pi r^2 = \frac{\pi r^2}{360}$.

 

Therefore, the area of the sector forming a degree measure of $\theta = \frac{\theta}{360} \times \pi r^2$.

 

Thus, $\text{The area of a sector} = \frac{\theta}{360} \times \pi r^2$, where $r$ is the centre of the circle and $\theta$ is the degree measure of the sector.

The length of the whole arc (or the circle) $=$ $2\pi r$

 

Thus, the length of an arc of a sector $=$ $\frac{\theta}{360} \times 2\pi r$.

Let us look at the image given below for a better understanding.

 

 

The area of the segment $APB$ $=$ The area of the sector $OAPB$ $-$ The area of \triangle $OAB$

 

$=$ $\frac{\theta}{360} \times \pi r^2$ $-$ The area of $\triangle OAB$

 

Also, $\text{The perimeter of the sector} = 2r + \frac{2 \pi r\theta}{360}$