Area of Part of a Circle

The area of a part of a circle is the amount of space enclosed in a segment of a circle, which is a region bounded by an arc of the circle and a chord connecting its endpoints.

Table of Contents:

• Area of Part of a Circle
• Solved Examples
• FAQs

Area of Part of a Circle

The area of a part of a circle is the amount of space enclosed in a segment of a circle, which is a region bounded by an arc of the circle and a chord connecting its endpoints.

The formula for the area of a part of a circle depends on the central angle that defines the segment.

If the central angle is in radians, the formula for the area of a segment of a circle is:

A = (1/2) r^2 (θ – sin θ)

where “A” is the area of the segment, “r” is the radius of the circle, and “θ” is the central angle in radians.

If the central angle is in degrees, the formula for the area of a segment of a circle is:

A = (1/2) r^2 [(π/180)θ – sin(θ(π/180))]

where “A” is the area of the segment, “r” is the radius of the circle, and “θ” is the central angle in degrees.

Note that these formulas assume that the segment is a minor segment, which means that the central angle is less than 180 degrees. If the segment is a major segment (central angle greater than 180 degrees), then the formula for the area of the segment is:

A = (1/2) r^2 (2π – θ – sin θ)

or

A = (1/2) r^2 [(2π/180) – θ – sin(θ(π/180))]

depending on whether the central angle is in radians or degrees.

In general, calculating the area of a part of a circle can be more complex than finding the area of a whole circle or a circular sector, so it’s important to carefully consider the geometry of the segment and choose the appropriate formula.

Area of Part of a Circle Solved Examples

Example 1: Find the area of a minor segment of a circle with radius 5 cm and central angle 60 degrees.

Solution:

First, we convert the central angle from degrees to radians:

60 degrees = (60/180)π radians = π/3 radians

Using the formula for the area of a minor segment in radians:

A = (1/2) r^2 (θ – sin θ)

A = (1/2) (5 cm)^2 ((π/3) – sin(π/3))

A ≈ 2.26 cm^2

Therefore, the area of the minor segment is approximately 2.26 square centimeters.

Example 2: Find the area of a major segment of a circle with radius 8 cm and central angle 210 degrees.

Solution:

First, we convert the central angle from degrees to radians:

210 degrees = (210/180)π radians = 7π/6 radians

Using the formula for the area of a major segment in radians:

A=(1/2) (8 cm)^2 ( 2π – 7π/6 – sin(7π/6 ))

A= 1/2 x 64 x 3.11

A= 32 x 3.11

A ≈ 99.77 square cm

Therefore, the area of the major segment is approximately 99.77 square centimeters.

Example 3: Find the area of a minor segment of a circle with radius 6 cm and central angle 45 degrees.

Solution:

First, we convert the central angle from degrees to radians:

45 degrees = (45/180)π radians = π/4 radians

Using the formula for the area of a minor segment in radians:

A = (1/2) r^2 (θ – sin θ)

A = (1/2) (6 cm)^2 ((π/4) – sin(π/4))

A ≈ 1.40 cm^2

Therefore, the area of the minor segment is approximately 1.40 square centimeters.

Area of Part of a Circle FAQS

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