What is a Sector of a Circle?(Definition, Formulas & Examples)

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

1. Find the perimeter of a sector with
   • radius $10.5$ cm and central angle $36^{\circ}$
   • radius $42$ cm and central angle $60^{\circ}$
2. 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

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)