A circle is a set of points in a plane that are equidistant from a given point. The distance from the centre is called the radius, and the point is called the centre. There are many parts in a circle such as radius, diameter, circumference, tangent, secant, chord, arc, segment, and sector.

Let’s understand what is a sector of a circle and what are the different formulas related to it.

## What is a Sector of a Circle?

A sector is a portion of a circle that 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. The shape of a sector of a circle looks like a pizza slice or a pie.

## Types of Sectors of a Circle

A circle containing a sector can be further divided into two regions.

- Minor Sector
- Major Sector

### Minor Sector

The smaller region formed in a circle containing a sector is known as a minor sector. The angle subtended by a minor sector at the centre of a circle (central angle) is less than $180^{\circ}$.

In the above figure $\text{AOBP}$ is a minor sector, with the central angle $\theta \lt 180^{\circ}$.

### Major Sector** **

The larger region formed in a circle containing a sector is known as a major sector. The angle subtended by a major sector at the centre of a circle (central angle) is greater than $180^{\circ}$.

In the above figure $\text{AOBQ}$ is a major sector, with the central angle $\theta \gt 180^{\circ}$.

**Note:** If $\theta$ is the measure of the central angle of a minor sector, then the measure of the central angle of the corresponding major sector is $360^{\circ} – \theta$.

## Perimeter of Sector

The perimeter of a figure is the boundary enclosing the figure.

In the above figure perimeter of sector $\text{OAPB} = \text{OA} + \text{APB} + \text{OB}$

$= r + \frac{\pi r \theta}{180} + r$

$= r \left(2 + \frac{\pi \theta}{180} \right)$

Thus the formula for perimeter of a sector is $r \left(2 + \frac{\pi \theta}{180} \right)$

where

$r$ is the radius of a circle

$\theta$ is the central angle (in degrees)

The formula for the perimeter of a sector when the central angle is measured in radians is $r \left(2+ \frac{\pi \theta \times \frac{180}{\pi}}{180} \right) = r(2 + \theta)$

### Examples on Perimeter of Sector

**Example 1:** Find the perimeter of a sector of a circle of radius $42$ cm subtending an angle of $60^{\circ}$.

Radius of circle $r = 42$ cm

Angle subtended by the sector $\theta = 60^{\circ}$

Area of sector is $r \left(2 + \frac{\pi \theta}{180} \right)$

$= 42 \times \left(2 + \frac{\pi \times 60}{180} \right)$

$= 42 \times \left(2 + \frac{\pi}{3} \right)$

$= 42 \times \left(\frac{6 + \pi}{3} \right)$

$= 14 \times (6 + \pi) = 14 \times \left(6 + \frac{22}{7} \right)$

$= 14 \times \frac{42 + 22}{7} = 2 \times 64 = 128$ cm

**Example 2:** Find the perimeter of a sector of a circle of radius $35$ cm subtending an angle of $72^{\circ}$.

Radius of circle $r = 42$ cm

Angle subtended by the sector $\theta = 60^{\circ}$

Area of sector is $r \left(2 + \frac{\pi \theta}{180} \right)$

$= 42 \times \left(2 + \frac{\pi \times 72}{180} \right)$

$= 42 \times \left(2 + \frac{\pi \times 2}{5} \right)$

$= 42 \times \frac{10 + 2 \pi }{5} = 136.8$ cm

## Area of Sector

The area of a sector is given by the formula $\text{Area } = \frac{\theta \pi r^2}{360}$, where $\theta$ is the central angle and $r$ is the radius of a circle.

The formula for the area of a sector when the central angle is measured in radians is $\frac{\theta \times \frac{180} {\pi} \times \pi r^2}{360} = \frac{\theta r^2}{2}$.

### Examples on Area of Sector

**Example 1:** For a given circle of radius $4$ cm, the angle of its sector is $45^{\circ}$. Find the area of the sector.

Radius of circle $r = 4$ cm

Angle of sector $\theta = 45^{\circ}$

Area of sector = $\frac{\theta \pi r^2}{360} = \frac{45 \times \frac{22}{7} \times 4^2}{360}$

$= \frac{\frac{22}{7} \times 16}{8} = \frac{22}{7} \times 2 = \frac{44}{7} = 6.29 \text{ cm}^2$.

**Example 2:** Find the area of the sector with a central angle of $30^{\circ}$ and a radius of $9$ cm.

Radius of circle $r = 9$ cm

Angle of sector $\theta = 30^{\circ}$

Area of sector = $\frac{\theta \pi r^2}{360} = \frac{30 \times \frac{22}{7} \times 9^2}{360}$

$= \frac{\frac{22}{7} \times 81}{12} = 21.21 \text{ cm}^2$

**Example 3:** Find the area of a sector if the radius of the circle is $6$ cm, and the angle subtended at the centre = $\frac{2 \pi}{3}$.

Radius of circle $r = 6$ cm

Angle of sector $\theta = \frac{2 \pi}{3}$

Area of sector = $\frac{\theta r^2}{2} = \frac{\frac{2 \pi}{3} \times 6^2}{2} = \frac{\frac{2 \pi}{3} \times 36}{2}$

$=\frac{\frac{2 \pi}{3} \times 36}{2} = 12 \pi$

$= 12 \times \frac{22}{7} = 37.71 \text{ cm}^2$

## Practice Problems

- Find the perimeter of a sector with
- radius $10.5$ cm and central angle $36^{\circ}$
- radius $42$ cm and central angle $60^{\circ}$

- Find the area of a sector with
- radius $21$ cm and central angle $72^{\circ}$
- radius $4.9$ cm and central angle $30^{\circ}$

## FAQs

### What is the formula for the area of a sector of a circle?

Area of a sector of a circle = $\frac{\theta \times r^2 }{2}$ where $\theta$ is measured in radians. The formula can also be represented as Sector Area = $\frac{\theta}{360} \pi r^2$, where $\theta$ is measured in degrees.

### What do you understand by the sector of a circle?

The part of a circle covered by $2$ radii of a circle and their intercepted arc(the arc coming in that portion) is a sector of a circle. It is also known by the term pie-shaped part of a circle.

### What is a perimeter of a sector of a circle?

The total length of the circumference of the circle extends within the angle $\theta$ is a perimeter of a sector of a circle or in other words the sum of the lengths of the arc and the two radii. The formula to calculate the perimeter of a sector of a circle is $r \left(2 + \frac{\pi \theta}{180} \right)$.

### What is the difference between a sector and an arc?

An arc is a fraction of the circumference and part of a circle whereas a sector is a pie-shaped part of a circle covered with 2 radii.

### How many types of sectors are there in a circle?

There are two sectors in a circle. If the circle is divided into two equal portions that are in semicircles then the sectors are of the same size otherwise in other cases, if part of a circle is pie-shaped then one sector is larger than the other. The larger one is known as the major sector and the smaller one is known as a minor sector of a circle.

## Conclusion

A sector is a portion of a circle that is enclosed between its two radii and the arc adjoining them. The perimeter of a sector is calculated using a formula $r \left(2 + \frac{\pi \theta}{180} \right)$ and the area is calculated using a formula $\frac{\theta}{360} \pi r^2$, where $r$ is the radius of a circle and $\theta$ is the central angle of a sector.

## Recommended Reading

- Reference Books
- Sample Papers
- How to Construct a Tangent to a Circle(With Steps & Pictures)
- Angles in a Circle – Meaning, Properties & Examples
- Tangent of a Circle – Meaning, Properties, Examples
- How to Draw a Circle(With Steps & Pictures)
- Chord of a Circle – Definition, Properties & Examples
- What is a Circle – Parts, Properties & Examples
- Area of a Circle – Formula, Derivation & Examples
- Circumference (Perimeter) of a Circle – Definition, Formula & Examples
- What is an Arc of a Circle?(Definition, Formulas & Examples)