# Sector of a Circle: Definition, Formula, Area, Perimeter, Examples

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## What Is a Sector of a Circle?

A sector of a circle is a portion or part of a circle that is composed of an arc and its two radii. You can compare the sector of a circle to the shape of a pizza slice. A sector is formed when two radii of the circle meet at both ends of the arc. An arc is simply a portion of the circumference of the circle.

### Sector of a Circle Definition

The definition of the sector of a circle in geometry can be given as the part of the circle enclosed by two radii and an arc of the circle. The arc of the circle is a part of the boundary/circumference of the circle.

Two radii meet at the center of the circle to form two sectors.

• Minor sector
• Major sector

### Minor Sector

A sector of a circle is called the minor sector if the minor arc of the circle is a part of its boundary. It is the sector with a smaller area. The angle of a minor sector is less than 180 degrees.

### Major Sector

A sector is called the major sector if the major arc of the circle is a part of its boundary.

It is the sector with the greater area. The angle of a major sector is greater than 180 degrees.

## Sector of a Circle Formulas

Let’s learn how to find the area of a sector of a circle. The formula for determining the area of a sector is given in two ways, with an angle and without an angle.

### Area of a Sector Formula: When Angle Is Given

If the radius of a circle is given as “r” and the angle of the sector is given as . This angle is made by the two radii at the center.

As we know, for a complete circle, the angle made at the center is equal to 2 or $360^\circ$.

• If is measured in degrees, then “the area of a sector of a circle formula” is given by

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

• If is measured in radians, then “the area of a sector of a circle formula” is given by

Area of sector $= \frac{1}{2} \times \theta \times r^2$

• Perimeter of sector $=2r + \frac{\theta}{360} \times 2\pi r$

### Area of Sector Formula: When Angle Is Not Given

How can we find the area of a sector of a circle when the central angle is not given? Let’s find out.

If l is the length of the arc, r is the radius of the circle and θ is the angle subtended at the center, then the angle is expressed in terms of l and r as

$\theta = \frac{l}{r}$, where is in radians.

If the angle of the sector $=2\pi$, the area of the sector (full circle) is $r^2$.

Similarly, for the angle $= 1$, the area of the sector $= \frac{\pi r^2}{2\pi} = \frac{r^2}{2}$

Thus, for the angle , area of the sector $\theta = \frac{r^2}{2} = \frac{l}{r}\times\frac{r^2}{2} = \frac{lr}{2}$

• Area of the sector without an angle $= \frac{lr}{2}$.
• Perimeter of sector $= 2r + l$

### Arc Length of a Sector Formula

The length of the arc “l” of the sector with angle is given by:

$l = \frac{\theta}{360}\times2\pi r$  …when $\theta$ is given in degrees

$l = \theta r$ …when$\theta$ is given in radians

## Facts about Sector of a Circle

• A section or part of a circle involved by two radii with a central angle $90^\circ$ is called a quadrant.
• A section or part of a circle involved by two radii with a central angle of $180^\circ$ is called a semicircle.
• The combination of any two hands (minute hands and hour hands or hour hands and second hands or minute hands and second hands) of a circular analog clock form sectors.

## Conclusion

In this article, we learned about the sector of a circle, minor and major sector, the sector formula for area, perimeter and arc length with and without angle. Now, let us look at some solved examples and practice questions.

## Solved Examples On Sector of a Circle

1. Calculate the area of the sector.

Solution:

The radius of sector $= r = 6$ inches

Angle of sector $= \theta = 60^\circ$

The area of sector $= \frac{\theta}{360^\circ}\times\pi r^2 = \frac{60^\circ}{360^\circ}\times3.14\times6^2 = 18.84$ sq. in.

1. Find the area of a sector of a circular region whose central angle is 3 radians with a radius of 5 feet.

Solution:

The radius of sector $= r = 5$ feet

Angle of sector $= \theta = 3$ radians

If is measured in radians, then

The area of the sector $= \frac{\theta}{2}\times r^2 = \frac{3}{2}\times5^2 =37.5$ sq. feet.

1. Find the central angle of a sector (in degrees) which has a 25 sq. yard area and a radius of 4 yards. Use $\pi = 3.14$.

Solution:

Radius of sector $= r = 4$ yards

Area of sector $= 25$ sq. yards

If is measured in degrees, then

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

$25 = \frac{\theta}{360^\circ}\times3.14\times4^2$

$\theta = \frac{25\times360}{3.14\times4^2} = 179.14^0$

1. Find the perimeter of the sector shown below.

Solution:

The radius of sector $= r = 8$ inches

Angle of sector $= \theta = 115^\circ$

The perimeter of sector $= 2r + \frac{\theta}{360}\times2\pi r$

$=(2\times8) + \frac{115^\circ}{360^\circ}\times(2\times3.14\times8)$

$=16 + (\frac{1}{4}\times23.14\times8)$

$=16 + 12.56$

$=28.56$ in

1. Find the area and perimeter of a sector with a radius of 10 feet and an arc length of 12.56 feet.

Solution:

The radius of sector $= r = 10$ feet

Arc length $= l = 12.56$ feet

Area of the sector without an angle $= \frac{lr}{2} = \frac{12.56\times10}{2}=62.8$ sq. feet

Perimeter of sector $= 2r + l = 2(10) + 12.56 = 32.56$ feet.

1. Find the arc length of a sector having a radius of 5 feet and a central angle of $120^\circ$.

Solution:

The radius of sector $= r = 5$ feet

Angle of sector $= \theta = 120^\circ$

The length of the arc “l” of the sector with angle is given by;

$l = \frac{\theta \pi r}{180} = \frac{120\times3.14\times5}{180} = 10.47$ feet

## Practice Problems On Sector of a Circle

1

### The sector of a circle is formed by two ____ and an arc.

chords
diameters
tangents
CorrectIncorrect
The sector of a circle is formed by two radii and an arc.
2

### The central angle of the minor sector is _________.

less than 270 degrees
less than 180 degrees
more than 270 degrees
more than 180 degrees
CorrectIncorrect
Correct answer is: less than 180 degrees
The central angle of the minor sector is less than 180 degrees.
3

### For the quadrant of a circle, the central angle is _________.

$90^\circ$
$180^\circ$
$60^\circ$
$45^\circ$
CorrectIncorrect
Correct answer is: $90^\circ$
The quadrant of a circle can be a sector of a circle with a central angle of $90^\circ$.
4

### If$\theta$is measured in radians, then the formula for the area of a sector of a circle is equal to ____________.

Area of sector $=\frac{\theta}{180^\circ}\times\pi r^2$
Area of sector $=\frac{\theta}{360^\circ}\times\pi r^2$
Area of sector $=\theta r^2$
Area of sector $=\frac{\theta}{2}\times r^2$
CorrectIncorrect
Correct answer is: Area of sector $=\frac{\theta}{2}\times r^2$
If $\theta$ is measured in radians, then
Area of sector $= \frac{\theta}{2}\times r^2$
5

### The area of the quadrant of a circle is equal to _____ of that of the circle.

$\frac{1}{2}$
$\frac{1}{3}$
$\frac{1}{4}$
$\frac{1}{8}$
CorrectIncorrect
Correct answer is: $\frac{1}{4}$
The area of the quadrant of a circle is equal to one-fourth, i.e., $\frac{1}{4}$ of that of the circle.

## Frequently Asked Questions On Sector of a Circle

Arc represents the part of the circumference. The sector of a circle is a part of the circle that is enclosed by two radii and an arc of the circle as a part of its boundary.

The area of the sector of a circle is the area of the part of a circle composed of an arc and two radii.

The perimeter of a sector is formed by two radii and an arc.

Perimeter of the sector $= 2r + l = 2r + \frac{\theta}{360} \times 2\pi r$, where $r =$ radius of the circle, $l =$ arc length, $\theta =$ angle of the sector.

There are two types of a sector, which can be categorized as a minor or a major part.

Area of the sector without an angle $= \frac{lr}{2}$, where $r =$ radius of the circle, $l =$ arc length.

The arc length of a sector of a circle can be found using the formula:

$l = \frac{\theta}{360} \times 2\pi r$ or,

$l = \frac{\theta \pi r}{180}$. Where, $r =$ radius of the circle, $l =$ arc length, $\theta =$ angle of the sector.