PUMPA - SMART LEARNING

எங்கள் ஆசிரியர்களுடன் 1-ஆன்-1 ஆலோசனை நேரத்தைப் பெறுங்கள். டாப்பர் ஆவதற்கு நாங்கள் பயிற்சி அளிப்போம்

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

Area of a sector

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 an arc of a sector

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

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

Area of the segment \(APB\)

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

Important!

On considering the two figures given below, we come to the conclusions that follow.

**1.**Area of the major sector \(OAQB\) \(=\) \(\pi r^2\) \(–\) Area of the minor sector \(OAPB\)

**2.**Area of the major segment \(AQB\) \(=\) \(\pi r^2\) \(–\) Area of the minor segment \(APB\)