Areas of Sector and Segment of a Circle

A sector of a circle is the region enclosed by two radii of the circle and the arc lying between them. It represents a portion of the circle formed by a central angle at the center.

Area of Sector

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

• The formula for the area of a 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

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

Area of Major Sector = Area of Circle - Area of Minor Sector

Segment

A segment of a circle is the region enclosed by a chord of the circle and the arc corresponding to that chord. It represents the area cut off from the circle by drawing a chord.

Area of Segment

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

Area of Segment = Area of Sector - Area of Triangle

• The formula for the area of a 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]

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

Area of Major Segment = Area of Circle - Area of Minor Segment

Real-Life Example of the Area of a Sector of a Circle

A slice of pizza is a real-life example of a sector of a circle because it is formed by two radii and the arc between them.

Suppose a pizza of radius 7 inches is divided into 6 equal slices.

Each slice has a central angle of:

360° ÷ 6 = 60°

Using the sector area formula:

Area = (θ / 360°) × πr²
= (60° / 360°) × (22/7) × 7²
= 1/6 × 154
= 77/3
≈ 25.67 square inches

So, the area of each pizza slice is approximately 25.67 square inches.

Solved Examples

Example 1: Find the area of the major segment if the area of the minor segment is 4 cm² and the area of the circle is 10 cm².

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

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

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

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

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

Practice Problems

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

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

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

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