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Circle- A circle is a geometrical shape which is made up of an infinite number of points in a plane

that are located at a fixed distance from a point called as the center of the circle.

The fixed distance from any of these points to the center is known as the radius of the circle.

Sectors in circle- A sector is a portion of a circle which is enclosed between its two radii and the

arc joining them.

Like- a semi circle is also a sector with two radii as its diameter. It divides the whole circle in two equal parts or sectors. In this sector the angle between two radii is $180^\circ $.

But there are two types of sectors.

Major sector

Minor sector

The smaller area is known as the minor sector and the larger being the major sector.

In the diagram, $\theta $ is the central angle and “r” is the radius of the circle.

Here PAQO is a minor sector and PRQO is a major sector made by radius “r” and angle $\theta $.

Now in the next step we will learn how to calculate the area of sectors.

Step by step solution :

Keeping in mind that area of circle which is an angle of $360^\circ $ is $\pi {r^2}$.

We will calculate the area of sectors.

In the above circle with center O and radius r, let PAQO be a sector and $\theta $ (in degrees) be the angle of the sector.

When the angle is of $360^\circ $ area of the sector is $\pi {r^2}$.

So area of $1^\circ $ will be $\pi {r^2}/360^\circ $

therefore for the sector of angle $\theta $ area will be $\left( {\pi {r^2}/360^\circ } \right) \times \theta $

Or,

Area of minor sector PAQO is $ = \theta \times \pi {r^2}$

Example radius

4 units, the angle of its sector is \[45^\circ \]

Area of sector $ = \dfrac{\theta }{{360^\circ }} \times \pi {r^2}$

$ = \dfrac{{45^\circ }}{{360^\circ }} \times \pi {r^2}$

$ = 0.125 \times 3.14 \times {4^2} = 6.28$ sq. units

Area of sector $ = \dfrac{\theta }{{360^\circ }} \times \pi {r^2}$

$ = \dfrac{{180^\circ }}{{360^\circ }} \times \pi {r^2} = \dfrac{{\pi {r^2}}}{2}$

2. We can also calculate Area of major sector by subtracting area of minor sector from area of circle. i.e., $\pi {r^2} - \dfrac{\theta }{{360^\circ }} \times \pi {r^2}$