Area of Sector of a Circle – Definition, Formula, Examples, FAQs

Home » Math Vocabulary » Area of Sector of a Circle – Definition, Formula, Examples, FAQs

What Is the Area of a Sector?

The area of a sector is the area of the region enclosed by an arc and two radii of a circle. It represents a part of the area of a circle. Area of a sector is measured in square units, depending on the unit of the radius.

Sector of a circle

What Is the Sector of a Circle?

A sector is a part of a circle made of the arc of the circle along with its two radii. In the given diagram, the yellow region represents the sector of the circle. It is formed by two radii and an arc. The angle is the angle made by the sector at the center. 

Area of a Sector: Formulas

1) The formula to calculate the area of a sector of a circle when θ is in degrees is given by:

Area  of a sector =θ360×πr2

where:

  • θ is the angle of the sector in degrees (angle subtended by the arc at the center)
  • r is the radius of the circle
Area of sector when central angle is in degrees

(Note: In the above diagram, the term ‘degrees’ simply signifies that this formula for the area of a sector is to be used when the central angle is given in degrees.)

2) The formula to calculate the area of a sector of a circle when θ is in radians is given by:

Area  of a sector =(θ2)×r2

where:

  • θ is the angle of the sector in radians
  • r is the radius of the circle
Area of sector formula in radians

(Note: In the above diagram, the term ‘radians’ simply signifies that this formula for the area of a sector is to be used when the central angle is in radians.)

How to Calculate the Area of a Sector

Consider these steps to determine a sector’s area:

Step 1: Note down the radius and the central angle () of the sector.

Step 2: If the angle is in degrees, then substitute the values in the formula

Area =(θ360)×πr2

Step 3: If the angle is in radians, then substitute the values in the formula

Area =(θ2)×r2

Step 4: The area is measured in square units. Assign the appropriate unit.

Area of a Sector Formula: Derivation

Area of a circle and area of sector

When angle is in degrees:

The angle made by a complete circle around the center is θ=360.

Area of circle with radius r=πr2

There’s a direct relationship between the central angle and area.

Central angle ()Area 
360πr2
θA=?

AreaofasectorAreaofacircle=CentralAngle360

Aπr2=360

A=θ360×πr2

When angle is in radians:

The angle made by a complete circle around the center is θ=2π.

Area of circle with radius r=πr2

Central angle ()Area 
2ππr2
θA = ?

AreaofasectorAreaofacircle=CentralAngle2π

Aπr2=θ2π

A=θ2π×πr2

A=θ2×r2

Area of a Sector in Degrees

The area of a sector when the central angle (θ) is expressed in degrees, is given by

Area=(θ360)×πr2

Example: Find the area of the given sector.

Sector with angle 45 degrees and radius 5 units

r=5 units

=45

Area =(θ360)×πr2

Area =(45360)×π(5)2

Area =18×3.14×25

Area =9.8125 square units

Area of a Sector in Radians

The area of a sector when the central angle (θ) is expressed in degrees, is given by

Area =(θ2)×r2

Example: Find the area of the sector of the circle. (Use π=3.14).

Sector with angle /4 degrees and radius 7 units

Area =(π4×2)×(7)2

Area =(3.144×2)×(7)2

Area = 19.2325 square units

Facts about Area of a Sector

  • The area of a sector is directly proportional to the central angle. The area of a sector increases as the central angle increases.
  • The area of a sector is always proportional to the square of the radius.
  • The maximum area a sector can have is when the central angle is 360 degrees, which corresponds to the entire circle. In this case, the area of the sector is equal to the area of the whole circle. Thus, a circle is a sector with a central angle of 360 degrees and area of 
  • πr2. 
  • A semicircle is a sector with a central angle of 180 degrees. Its area if half the area of a circle, given by πr22.

Conclusion

In this article, we learned about the area of a sector, its formulas, and also the derivation of the formulas. Let’s solve a few examples and MCQs for practice.

Solved Examples on Area of a Sector

Example 1: Find the area of the sector in terms of π if the circle has a radius of 8 units and a central angle of 45 degrees.

Solution: 

Using the formula for sector area, we write 

Area  =θ360×πr2

Here, r=8 units, θ=45

Area =45360×π×82

Area =(18)×π×64

Area =8π square units

Example 2: A sector has a radius of 12 units and a central angle of 90 degrees. Calculate the area of the sector.

Solution: 

Area  =θ360×πr2

Here, r=12 units, θ=90

Area =90360×π×122

Area =14×π×144

Area =36π

Area = 113.04 square units

Example 3: A sector has an area of 16π square units and a radius of 4 units. Find the central angle of the sector in radians.

Solution: 

 r=4 units

Area of sector =16π square units

Area of sector =(θ2)×r

16π=(θ2)×(4)2

16π=(θ2)×16

\frac{θ}{2} = π$

θ=2π

Central angle is 2π. It means that the given area is the area of the whole circle with radius 4 units.

Example 4: Find the area of a sector with a central angle of 60 degrees and a radius of 10 units.

Solution: 

Area =(θ360)×πr2

Area =(60360)×π×102 

Area =(16)×π×100

Area =52.33 square units

Practice Problems on Area of a Sector

Area of Sector of a Circle - Definition, Formula, Examples, FAQs

Attend this quiz & Test your knowledge.

1

A sector has a radius of 12 units and a central angle of 60 degrees. What is the area of the sector?

144π cm²
72π cm²
36π cm²
18π cm²
CorrectIncorrect
Correct answer is: 72π cm²
Area =θ360×πr2
Area =60360×π×144
Area =24π square units
2

The area of a sector when the central angle is given in degrees is given by

Area =θ360×2πr
Area =θ90×πr2
Area =θ360×πr2
Area =θ180×πr2
CorrectIncorrect
Correct answer is: Area =θ360×πr2
Area of sector =θ360πr2
3

A sector has an area of 8π square units and a radius of 4 units. What is the central angle of the sector?

30 degrees
45 degrees
60 degrees
180 degrees
CorrectIncorrect
Correct answer is: 180 degrees
Area of sector =θ360×πr2
8π=θ360×π×16
12=θ360
θ=180
4

The area of a quarter circle is ___ times the area of a circle.

12
four
14
two
CorrectIncorrect
Correct answer is: 14
The area of a quarter circle is 14 times the area of a circle.

Frequently Asked Questions on Area of a Sector

A sector’s area is a fraction of the circle’s overall area. It is the area that the circle’s arc and two radii enclose. On the other hand, the term “area of a sector of a circle” describes the entire region that the circle’s circumference encloses.

No, a sector cannot have a larger area than the complete circle. Always a piece or proportion of the entire circle, a sector. As a result, its surface area is never greater than or equal to the circles.

The area of sector formula is produced by considering the percentage of the circle’s angle that it encloses. It is calculated by dividing the angle’s fraction by the circle’s whole area. This relationship is denoted by the formula  Asec=(θ360)×π×r2, where the sector’s central angle and r is the circle’s diameter.

A sector’s area cannot be negative. A region’s area defines its size, which is always a positive number. Area is a non-negative quantity.