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.
Table of Content
- What is Sector and Segment of a Circle?
- Areas of Sector and Segment of a Circle
- Examples on Areas of Sector and Segment of a Circle
- Practice Problems on Areas of Sector and Segment of a Circle
- FAQs on Areas of Segment and Sector 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 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.
Contact Us