Areas of Sector and Segment of a Circle

Areas of sector and segment of a circle with radius r and subtends an angle θ (in radians) are given by (1/2)×θr² and (1/2)×r²(θ -sinθ) respectively. The area of the sector and the area of the segment of the circle are easily calculated using the above formula.

In this article, we will explore the areas of sector and segment in detail and also learn the basics of sector and segment of a circle.

What is Sector and Segment of a Circle?

Sector of a circle is the region inside the circle made by two radii and the arc of the circle connecting the two radii of the circle. The segment of a circle is the region inside the circle made by the chord and the arc connecting the two endpoints of the chord.

Definition of Sector

The region formed by the two radii of the circle and the arc between them is called the sector of a circle. The sector of a circle can be of two types:

• Major Sector
• Minor Sector

The diagram below represents the sector of a circle.

Definition of Sector

Definition of Segment

The region formed by the chord of circle and the arc between the two points of the chord is called as segment of a circle. The segment of circle can be of two types:

• Major Segment
• Minor Segment

The below diagram represents the segment of the circle.

Areas of Sector and Segment of a Circle

Below we will discuss the area of sector as well as the area of segment of a circle.

Area of Sector

Area of sector of a circle is determined by multiplying angle subtended by the sector and area of the circle and further dividing the result with 360°.

Formula for Area of Sector of a Circle

Formula for area of sector is given by:

Area of Sector (when θ is in degrees) = πr² × (θ / 360°)

Area of Sector (when θ is in radians) = (1/2) × θr²

Formula for Area of Major Sector of a Circle

Formula for the area of major sector of a circle is given by:

Area of Major Sector = Area of Circle – Area of Minor Sector

Area of Segment

Area of segment of a circle is given by subtracting the area of triangle from the area of the sector. From the figure below we can clearly see that the area of segment of circle is equal to the difference of area of sector and area of triangle.

Area of Segment = Area of Sector – Area of Triangle

Formula for Area of Segment of a Circle

Formula for the area of segment of a circle is given below:

Area of Segment (when θ is in radians) = (1/2) × r²(θ – sinθ)

Area of Segment (when θ is in degrees) = (1/2) × r²[(π/180)θ – sinθ]

Formula for Area of Major Segment of a Circle

Formula for the area of major segment of a circle is given by:

Area of Major Segment = Area of Circle – Area of Minor Segment

Examples on Areas of Sector and Segment of a Circle

Example 1: Find the area of the sector given that radius of circle is 4 cm and angle subtended by sector is π/3 radians.

Solution:

To find area of sector we use following formula

Area of Sector = (1/2) × θr²

Area of Sector = (1/2) × (π/3)4²

Area of Sector = 8 × (π/3)

Area of Sector = 8.38 cm²

Example 2: Determine the area of segment given the radius of the circle is 2 cm and angle subtended by segment is 90°.

Solution:

To find area of segment we use following formula

Area of Segment (when θ is in degrees) = (1/2) × r²[(π/180)θ – sinθ]

Area of Segment = (1/2) × 2²[(π/180°)90° – sin90°]

Area of Segment = (1/2) × 4[(π/2) – 1]

Area of Segment = 2 × [(π/2) – 1]

Area of Segment = 1.142 cm²

Example 3: Find the area of the major segment if the area of minor segment is 4 cm² and area of circle is 10 cm².

Solution:

To find area of major segment we use formula

Area of Major Segment = Area of Circle – Area of Minor Segment

Area of Major Segment = 10 – 4

Area of Major Segment = 6 cm²

Example 4: Determine the area of the minor sector if the area of major sector is 110 cm² and area of circle is 200 cm².

Soliution:

To find area of minor sector we use formula

Area of Major Sector = Area of Circle – Area of Minor Sector

Area of Minor Sector = 200 – 110

Area of Minor Sector = 90 cm²

Practice Problems on Areas of Sector and Segment of a Circle

Q1: Find the area of the sector given that radius of circle is 15 cm and angle subtended by sector is 60°.

Q2: Determine the area of segment given the radius of the circle is 27 cm and angle subtended by segment is π/3 radians.

Q3: Find the area of the major segment if the area of minor segment is 10 cm² and area of circle is 30 cm².

Q4: Determine the area of the minor sector if the area of major sector is 70 cm² and area of circle is 120 cm².

FAQs on Areas of Segment and Sector of a Circle

What is Formula for Area of Sector of Circle?

The formula for area of a circle sector is given by:

• Area of Sector (when θ is in degrees) = πr² × (θ / 360°)

• Area of Sector (when θ is in radians) = (1/2) × θr²

What is Formula for Area of Segment of Circle?

The formula for the area of a circle segment is given by:

• Area of Segment (when θ is in radians) = (1/2) × r²(θ – sinθ)

• Area of Segment (when θ is in degrees) = (1/2) × r²[(π/180)θ – sinθ]

What is an Example of a Segment and a Sector of a Circle?

An example of segment of circle is semicircle which is the largest segment of a circle whereas an example of sector of circle is a pizza slice.

What is Difference Between a Segment and a Sector in a Circle?

Difference between a segment and a sector in a circle is that the segment is made by the chord and arc joining the end points of the chord whereas the sector is made by the two radii and arc joining the two radii.