102 LearnersLast updated on July 29th, 2025
Perimeter of Arc
The perimeter of a shape is the total length of its boundary. The perimeter of an arc refers to the length of the curved part of a circle, which can be used in designing structures, gardens, and more. In this topic, we will learn about the perimeter of an arc.
What is the Perimeter of an Arc?
The perimeter of an arc is the length of the curved boundary of a part of a circle. It is calculated as a fraction of the circumference of the circle, which is proportional to the angle that the arc subtends at the center of the circle. The formula for the perimeter of an arc is 𝑃 = 2πr(θ/360) + 2r, where r is the radius of the circle and θ is the angle in degrees. For instance, if a circle has a radius of 5 units and the arc subtends an angle of 60 degrees, then its perimeter is 𝑃 = 2π(5)(60/360) + 2(5) = 5π/3 + 10.
Formula for Perimeter of Arc - 𝑷 = 2πr(θ/360) + 2r.
Let’s consider another example of calculating the perimeter of an arc with a circle of radius 7 units and an angle of 90 degrees. The perimeter of the arc will be: 𝑃 = 2π(7)(90/360) + 2(7) = 7π/2 + 14.
How to Calculate the Perimeter of an Arc
To find the perimeter of an arc, apply the given formula and calculate both the arc length and the two radii. For instance, if a circle has a radius of 8 units and the arc subtends an angle of 45 degrees, then: Arc Length = 2π(8)(45/360) = π/2 Perimeter = Arc Length + 2(8) = π/2 + 16 units. Example Problem on Perimeter of Arc - For finding the perimeter of an arc, we use the formula, 𝑷 = 2πr(θ/360) + 2r. For example, let’s say, r = 6 units and θ = 120 degrees. Now, the perimeter = Arc Length + 2r = 2π(6)(120/360) + 2(6) = 4π + 12. Therefore, the perimeter of the arc is 4π + 12 units.
Tips and Tricks for Perimeter of Arc
Learning some tips and tricks makes it easier for children to calculate the perimeter of arcs. Here are some tips and tricks given below: Always remember that an arc's perimeter involves both the arc length and the two radii of the circle. Use the formula, 𝑷 = 2πr(θ/360) + 2r. Calculating the arc length starts by determining the radius and the angle. Ensure the angle is in degrees for using the formula. To reduce confusion, arrange the given measurements clearly when dealing with multiple arcs. Apply the formula to each arc separately. To avoid mistakes when calculating the perimeter, ensure the radius and angle measurements are accurate and consistent, especially for practical uses like construction and decoration. If you are given the arc length and need to find the perimeter, remember to add twice the radius to the arc length.
Common Mistakes and How to Avoid Them in Perimeter of Arc
Did you know that while working with the perimeter of an arc, children might encounter some errors or difficulties? We have many solutions to resolve these problems. Here are some given below:
Mistake 1
Incorrectly using the full circumference formula for arcs.
Remember that the arc length is only a fraction of the full circumference. Use the formula 2πr(θ/360) to find the arc length.
Mistake 2
Confusing radians and degrees, leading to incorrect calculations.
Ensure that the angle used in the formula is in degrees, or convert radians to degrees before calculating.
Mistake 3
Assuming all arcs are semicircles or quarter circles, which results in inaccurate calculations.
Arcs can represent any fraction of a circle. Use the given angle to determine the appropriate arc length.
Mistake 4
Using the wrong values for the radius or angle, misreading a problem, or mislabeling the measurements.
Double-check the problem and ensure that you're using the correct radius and angle for your arc.
Mistake 5
Misunderstanding or misidentifying the angle measurement given in a problem outline.
Before starting the calculations, make sure that you understand the problem and, if possible, visualize the circle to accurately identify the angle involved.
Perimeter of Arc Examples
Problem 1
A circular garden has an arc with a perimeter of 40 units. The radius of the garden is 10 units, and the arc subtends an angle of 72 degrees. Calculate the arc length.
Arc Length = 12.57 units (approximately).
Explanation
Let ‘L’ be the arc length. And the given perimeter = 40 units. Radius of the garden = 10 units. Perimeter of arc = Arc Length + 2(radius). 40 = L + 2(10) 40 = L + 20 L = 40 – 20 = 20 Therefore, the arc length is approximately 12.57 units.
Problem 2
A wire of length 150 units is used to form an arc of a circle with a radius of 15 units. The arc subtends an angle of 180 degrees at the center. Find the perimeter of this arc.
Perimeter = 60π + 30 units.
Explanation
Given that the length of the wire forms the arc, the perimeter is calculated as: Arc Length = 2π(15)(180/360) = 15π Perimeter of arc = Arc Length + 2(radius) = 15π + 2(15) = 15π + 30 Therefore, the perimeter of the arc is 15π + 30 units.
Problem 3
Find the perimeter of an arc of a circle with a radius of 5 units and an angle of 45 degrees.
Perimeter = (5π/4) + 10 units.
Explanation
Perimeter of arc = Arc Length + 2(radius) = 2π(5)(45/360) + 2(5) = (5π/4) + 10 Therefore, the perimeter of the arc is (5π/4) + 10 units.
Problem 4
An artist is designing a circular mosaic with an arc. The circle has a radius of 12 units, and the arc subtends an angle of 60 degrees. How much material is needed to cover the perimeter of the arc?
The artist will need 4π + 24 units of material.
Explanation
The perimeter of the arc includes the arc length and the two radii. Using the formula: Arc Length = 2π(12)(60/360) = 4π Perimeter = Arc Length + 2(radius) = 4π + 2(12) = 4π + 24 units.
Problem 5
Find the perimeter of the arc of a circle with radius 9 units and an angle of 120 degrees.
Perimeter = 3π + 18 units.
Explanation
The perimeter of the arc includes the arc length and the two radii. The entire distance is calculated as: Arc Length = 2π(9)(120/360) = 3π Perimeter = Arc Length + 2(radius) = 3π + 18 units.
FAQs on Perimeter of Arc
1.Evaluate the arc’s perimeter if its radius is 6 units and the angle is 30 degrees.
2.What is meant by an arc’s perimeter?
3.What are the types of arcs?
4.Which arc represents half the circle?
5.Which angle results in the smallest arc?
Important Glossaries for Perimeter of Arc
Arc: A part of the circumference of a circle. Radius: The distance from the center of a circle to any point on its boundary. Angle: The measure in degrees of the circle's sector that the arc spans. Circumference: The total length around a circle. Perimeter of Arc: The sum of the arc length and twice the radius.
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
