Perimeter of Segment
2026-02-28 10:45 Diff
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Last updated on August 5, 2025

The perimeter of a shape is the total length of its boundary. The perimeter of a segment, specifically, is the sum of the arc length and the lengths of both radii connecting the endpoints of the arc to the center of the circle. Perimeter is also used in various practical applications, such as designing paths, creating boundaries, and more. In this topic, we will learn about the perimeter of a segment.

What is the Perimeter of a Segment?

The perimeter of a segment is the total length along the arc and the two radii. By adding the arc length and the lengths of both radii, we get the perimeter of the segment. The formula for the perimeter of a segment is 𝑃 = 𝑟𝜃 + 2𝑟, where 𝑟 is the radius and 𝜃 is the angle (in radians) subtended by the arc at the center of the circle. For instance, if a segment has a radius 𝑟 = 6 and an angle 𝜃 = 1 radian, then its perimeter is 𝑃 = 6 × 1 + 2 × 6 = 18.

Formula for Perimeter of Segment - 𝑃 = 𝑟𝜃 + 2𝑟.

Let’s consider another example of a segment with a radius 𝑟 = 8 and an angle 𝜃 = 1.5 radians. So the perimeter of the segment will be: 𝑃 = 𝑟𝜃 + 2𝑟 = 8 × 1.5 + 2 × 8 = 28.

How to Calculate the Perimeter of a Segment

To find the perimeter of a segment, we just need to apply the given formula and sum the arc length and the lengths of the radii. For instance, a given segment has a radius 𝑟 = 6 and an angle 𝜃 = 2 radians. Perimeter = arc length + sum of radii = 6 × 2 + 2 × 6 = 24 cm. Example Problem on Perimeter of Segment - For finding the perimeter of a segment, we use the formula, 𝑃 = 𝑟𝜃 + 2𝑟. For example, let’s say, 𝑟 = 5 cm and 𝜃 = 1.2 radians. Now, the perimeter = arc length + sum of radii = 5 × 1.2 + 2 × 5 = 16 cm. Therefore, the perimeter of the segment is 16 cm.

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Tips and Tricks for Perimeter of Segment

Learning some tips and tricks makes it easier for children to calculate the perimeter of segments. Here are some tips and tricks given below: Always remember that a segment's perimeter is the sum of the arc length and the two radii. For that, use the formula, 𝑃 = 𝑟𝜃 + 2𝑟. Calculating the perimeter of a segment starts by determining the arc length using the angle in radians. The formula for the arc length is: Arc Length = 𝑟𝜃. To reduce confusion, specifically arrange the given measurements if you need the perimeter of a group of segments. After that, apply the formula to each segment. To avoid mistakes when adding the perimeter, make sure the measurements are precise and constant for common uses like landscaping and architecture. If you are given the central angle in degrees, convert it to radians before calculating the arc length using the formula: Radians = Degrees × (π/180).

Common Mistakes and How to Avoid Them in Perimeter of Segment

Did you know that while working with the perimeter of a segment, children might encounter some errors or difficulties? We have many solutions to resolve these problems. Here are some given below:

Problem 1

A circular park has a segment with a radius of 10 meters and a central angle of 1.5 radians. Calculate the perimeter of this segment.

Okay, lets begin

The perimeter of the segment is 35 meters.

Explanation

Let 𝑟 be the radius and 𝜃 be the central angle. Given radius, 𝑟 = 10 meters, and angle, 𝜃 = 1.5 radians. Perimeter of the segment = arc length + sum of radii. Arc length = 𝑟𝜃 = 10 × 1.5 = 15 meters. Perimeter = 15 + 2 × 10 = 35 meters.

Well explained 👍

Problem 2

A wire of length 150 inches is bent into a circular segment with a radius of 20 inches and an angle of 2 radians. Find the remaining length of the wire after forming the segment.

Okay, lets begin

The remaining length of the wire is 90 inches.

Explanation

Given the total length of the wire is 150 inches. Radius of the segment, 𝑟 = 20 inches, and angle, 𝜃 = 2 radians. Perimeter of the segment = arc length + sum of radii. Arc length = 𝑟𝜃 = 20 × 2 = 40 inches. Perimeter = 40 + 2 × 20 = 80 inches. Remaining length of the wire = 150 - 80 = 70 inches.

Well explained 👍

Problem 3

Find the perimeter of a segment in a circle with a radius of 12 cm and an angle of π/3 radians.

Okay, lets begin

The perimeter of the segment is 32 cm.

Explanation

Perimeter of the segment = arc length + sum of radii. Arc length = 𝑟𝜃 = 12 × π/3 = 4π cm. Perimeter = 4π + 2 × 12 = 32 cm (approximately, using π ≈ 3.14).

Well explained 👍

Problem 4

Annie is designing a flower bed in the shape of a circular segment. The radius of the circle is 9 meters, and the central angle is 0.8 radians. How much fencing will Annie need to enclose the flower bed?

Okay, lets begin

Annie will need 25.2 meters of fencing.

Explanation

The perimeter of the segment is the sum of the arc length and the lengths of the two radii. Using the formula: Arc length = 𝑟𝜃 = 9 × 0.8 = 7.2 meters. Perimeter = 7.2 + 2 × 9 = 25.2 meters.

Well explained 👍

Problem 5

Find the perimeter of a segment formed by a circle with a radius of 7 meters and a central angle of 1.2 radians.

Okay, lets begin

The perimeter of the segment is 21.4 meters.

Explanation

Perimeter of the segment = arc length + sum of radii. Arc length = 𝑟𝜃 = 7 × 1.2 = 8.4 meters. Perimeter = 8.4 + 2 × 7 = 21.4 meters.

Well explained 👍

FAQs on Perimeter of Segment

1.Evaluate the segment’s perimeter if its radius is 5 cm and the angle is 2 radians.

Perimeter of the segment = arc length + sum of radii. Hence, 𝑃 = 5 × 2 + 2 × 5 = 20 cm.

2.What is meant by a segment’s perimeter?

The total length around a segment, including the arc and the two radii, is its perimeter.

3.What are the types of segments in a circle?

Segments can be classified based on the central angle: minor segment (central angle less than π) and major segment (central angle more than π).

4.Which segment has the smallest perimeter for a given radius?

For a given radius, the segment with the smallest central angle (close to zero) will have the smallest perimeter, as the arc length approaches zero.

5.How is the arc length of a segment calculated?

The arc length of a segment is calculated using the formula: Arc Length = 𝑟𝜃, where 𝑟 is the radius and 𝜃 is the central angle in radians.

Important Glossaries for Perimeter of Segment

Perimeter: The total length around a shape, including all its sides or edges. Segment: A part of a circle bounded by an arc and the radii joining the endpoints of the arc to the center. Arc Length: The distance along the curved line making up the arc, calculated as 𝑟𝜃. Radius: A straight line from the center of a circle to any point on its circumference. Central Angle: The angle subtended at the center of a circle by two radii.

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Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

Fun Fact

: She has songs for each table which helps her to remember the tables