2. Important formulae

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