**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.

A part of a curve lying on the circumference of a circle.

**Sector-**

A sector is a portion of a circle which is enclosed between its two radii and the arc adjoining them. The most common sector of a circle is a semi-circle which represents half of a circle.

A circle containing a sector can be further divided into two regions known as a Major Sector and a Minor Sector.

In the figure below,* OPBQ* is known as the **M****ajor Sector** and *OPAQ* is known as the **M****inor Sector. **As Major represent big or large and Minor represent Small, which is why they are known as Major and Minor Sector respectively. In a semi-circle, there is no major or minor sector.

We know that a full circle is 360 degrees in measurement. Area of a circle is given as π times the square of its radius length. So if a sector of any circle of radius r measures θ, area of the sector can be given by:

**Area of sector = \(\frac{\theta }{360} \times \pi r^{2}\)**

**Arc- **

A part of a curve lying on the circumference of a circle.

**Length of an arc of a sector- **The length of an arc is given as-

**\(\frac{\theta }{360} \times 2 \pi r\)**

There are instances where the angle of a sector might not be given to you. Instead, the length of the arc is known. In such cases, you can compute the area by making use of the following:

**Derivation:**

In a circle with center O and radius r, let OPAQ be a sector and θ (in degrees) be the angle of the sector.

Area of the circular region is πr².

Let this region be a sector forming an angle of 360° at the centre O.

Then, area of a sector of circle formula is calculated using the unitary method.

When the angle at the center is 360°, area of the sector, i.e., the complete circle = πr²

When the angle at the center is 1°, area of the sector = \(\frac{\pi .r ^{2}}{360^{0}}\)

Thus, when the angle is θ, area of sector, *OPAQ* = \(\frac{\theta }{360^{o}}\times \Pi r^{2}\)

**Solved Examples:**

**Questions 1****: For a given circle of radius 4 units, the angle of its sector is 45°. Find the area of the sector.**

**Solution:** Given, radius r = 4 units

Angle θ = 45°

Area of the sector = \(\frac{\theta }{360^{o}}\times \Pi r^{2}\)

= \(\frac{45}{360^{0}}\times\frac{22}{7}\times 4^{2}=6.28\;sq.units\)

**Questions ****2: Find the area of the sector with a central angle 30° and a radius of 9cm.**

**Solution:** Given,

Radius r = 9 cm

Angle θ = 30°

Area of the sector = \(\frac{\theta }{360^{0}}\times \Pi r^{2}\)

= \(\frac{30}{360^{0}}\times \frac{22}{7}\times 9^{2}=21.21cm^{2}\)

To solve more problems on the topic and for video lessons on the topic, download Byju’s -The Learning App.

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