Perimeter of a Sector Calculator
Welcome to our Perimeter of a Sector Calculator page.
We explain how to find the perimeter of a sector and provide a quick calculator to work it out for you step-by-step.
There are also some worked examples and worksheets to help you practice this skill.
Quicklinks to ...
Perimeter of a Sector Calculator
This calculator finds the perimeter of a sector (and also the arc length) when the radius (or diameter) and the angle (in degrees or radians) is known.
If the angle is more than one full turn (360° or 2π radians) then the perimeter and arc length will be calculated for this angle.
How the Perimeter of a Sector Calculator Works
- Type the angle you want for the sector, and choose whether to use degrees (default) or radians.
- Select if you want to use the radius or diameter (default is radius).
- Type the radius you want to use.
- For the radius and angle, you can choose a whole number, decimal or fraction.
- You can type a fraction by typing the numerator then '/' then the denominator.
- You can type a mixed number by typing the whole-number part, then a space then the fraction part.
- Examples: 2 1/2 (two and one-half); 3 4/5 (three and four-fifths); 7 1/3 (seven and one-third).
- Choose your units of measurement (default is none)
- Choose your desired accuracy (default is 2 decimal places)
- Click the Find Area button
- Depending on the angle and answer, you may be given two different answers, one in terms of Pi (π) and one just as a decimal answer.
What is a Sector of a Circle?
A sector is a section (or part) of a circle.
Sectors have central angles and radius values which we can use to find the area and perimeter.
In the image above you can see the light brown sector of the circle (called a minor sector) with a central angle of θ°
There is also a larger white sector (called a major sector) which has a central angle of (360 - θ)°
Sectors can have any central angle of up to 360°
The central angle can also be measured in radians - when it is measured in radians, the angle can be any value up to 2π radians.
When the central angle is measured in radians, the formula for both the area and perimeter is different.
Perimeter of a Sector Formula
To find the perimeter of a sector, we need to work out the distance around the outside of the sector which is radius + arc length + radius = 2 x radius + arc length.
The formula for the arc length is different depending on whether the angle is measured in degrees or radians.
The perimeter of a sector of a circle where the central angle is in degrees is: \[ P = 2r + L = 2r + {\theta \over 180} \pi r \]
The perimeter of a sector of a circle where the central angle is in radians is: \[ A = 2r + L = 2r + \theta r \]
where θ is the central angle and r is the radius of the circle.
Perimeter of a Sector Examples
Perimeter of a Sector Example 1
Find the perimeter of the sector below to 1 decimal place.
The sector has a central angle of 115° and a radius of 6 cm.
The perimeter of a sector:
\[ P = 2r + L \] \[ L = {\theta \over 180} \pi r \] where P is the perimeter, L is the arc length, θ is the central angle and r is the radius of the circle.
First, we are going to find the value of the arc length, L:
So if we substitute the values of the angle and radius into this equation, we get: \[ L = {115 \over 180} \pi \cdot 6 = {115 \cdot 6 \over 180} \pi = {690 \over 180} \pi \]
If we simplify this equation, we get: \[ L = {23 \over 6} \pi = 12.042 \; cm \; (to \; 3 \; dp) \]
We can now find the perimeter: \[ P = 2r + L = 2 \cdot 6 + 12.042 = 12 + 12.042 = 24.042 \; cm \]
This gives us a final answer of: \[ P = 24.0 \; cm \; to \; 1 \; decimal \; place \]
Perimeter of a Sector Example 2
Find the perimeter of the sector below to 1 decimal place.
The sector has a central angle of 72° and a radius of 8 ½ in.
The perimeter of a sector:
\[ P = 2r + L \] \[ L = {\theta \over 180} \pi r \] where P is the perimeter, L is the arc length, θ is the central angle and r is the radius of the circle.
First, we are going to find the value of the arc length, L:
So if we substitute the values of the angle and radius into this equation, we get: \[ L = {72 \over 180} \pi \cdot 8 {1 \over 2} = {72 \over 180} \pi \cdot {17 \over 2} \]
Multiplying this out gives us: \[ L = { 72 \cdot 17 \over 180 \cdot 2} \pi = {1224 \over 360} \]
If we simplify this equation, we get: \[ L = {17 \over 5} \pi = 10.681 \; in (to 3 dp) \]
We can now find the perimeter: \[ P = 2r + L = 2 \cdot 8 {1 \over 2} + 10.681 = 17 + 10.681 = 27.681 \; in \]
This gives us a final answer of: \[ P = 27.7 \; in \; to \; 1 \; decimal \; place \]
Perimeter of a Sector Example 3
Find the perimeter of the sector below, giving your answer to 1 decimal places.
The sector has a central angle of 278° and a radius of 9.6 cm.
The perimeter of a sector:
\[ P = 2r + L \] \[ L = {\theta \over 180} \pi r \] where P is the perimeter, L is the arc length, θ is the central angle and r is the radius of the circle.
First, we are going to find the value of the arc length L:
So if we substitute the values of the angle and radius into this equation, we get: \[ L = {278 \over 180} \pi \cdot 9.6 = {278 \cdot 9.6 \over 180} \pi \]
Multiplying this out gives us: \[ L = { 2668 \over 180} \pi = {1112 \over 75} \pi = 46.579 \; cm \]
We can now find the perimeter: \[ P = 2r + L = 2 \cdot 9.6 + 46.579 = 19.2 + 46.579 = 65.779 \; cm \]
This gives us a final answer of: \[ P = 65.8 \; cm \; to \; 1 \; decimal \; place \]
Perimeter of a Sector Example 4
Find the perimeter of the sector below, where the angle is in radians.
The sector has a central angle of 2 ½ radians and a radius of 16 m.
The perimeter of a sector, where the angle is in radians is:
\[ P = 2r + L \] \[ L = \theta r \] where P is the perimeter, L is the arc length, θ is the central angle and r is the radius of the circle.
First, we are going to find the value of the arc length L:
So if we substitute the values of the angle and radius into this equation, we get: \[ L = 2 {1 \over 2} \cdot 16 \]
Multiplying this out gives us: \[ L = { 5 \over 2} \cdot 16 = 40 \; m \]
We can now find the perimeter: \[ P = 2r + L = 2 \times 16 + 40 = 32 + 40 = 72 \; m \]
This gives us a final answer of: \[ P = 72 m \]
Perimeter of a Sector Example 5
Find the perimeter of the sector below, where the angle is in radians.
The sector has a central angle of 4.8 radians and a radius of 17 ½ inches.
The perimeter of a sector, where the angle is in radians is:
\[ P = 2r + L \] \[ L = \theta r \] where P is the perimeter, L is the arc length, θ is the central angle and r is the radius of the circle.
First, we are going to find the value of the arc length L:
So if we substitute the values of the angle and radius into this equation, we get: \[ L = 4.8 \cdot 17 {1 \over 2} \]
Simplifying this expression gives us: \[ L = 4.8 \cdot { 35 \over 2} = 84 \; in. \]
We can now find the perimeter: \[ P = 2r + L = 2 \times 17 {1 \over 2} + 84 = 35 + 84 = 119 \; in. \]
This gives us a final answer of: \[ P = 119 \; inches \]
Perimeter of a Sector Worksheets
We have created some worksheets to help you practice this skill.
The first and second sheets involve working out the perimeter of different sectors where the angle is measured in degrees.
Sheet 3 involves finding the perimeter of different sectors where the angle is measured in radians.
More Recommended Math Resources
Take a look at some more of our math resources similar to these.
Area of a Sector Calculator
We also have a calculator for finding the area of a sector using the angle and radius values.
Take a look at the webpage below where you will find formula, worked examples and a handy calculator for finding the area of sectors in degrees or radians.
More Circle Worksheets and Support Pages
Convert Radians to Degrees Calculator
Use this calculator for converting angles in radians to degrees.
There is also a handy formula and step-by-step instructions to do this conversion.
More Area & Volume Calculators
We have a range of area and volume calculators to help you find the area and volumes of a range of different 2d and 3d shapes.
Each calculator page comes with worked examples, formulas and practice worksheets.
Looking for more math calculators?
We have a wide range of free math calculators to help you.
Most of our calculators show you their working out so that you can see exactly what they have done to get the answer.
Our calculator hub page contains links to all of our calculators!
How to Print or Save these sheets
Need help with printing or saving?
Follow these 3 steps to get your worksheets printed perfectly!
How to Print or Save these sheets
Need help with printing or saving?
Follow these 3 steps to get your worksheets printed perfectly!
Return to Calculators Hub page
Return from Perimeter of a Sector Calculator to Math Salamanders Homepage
Math-Salamanders.com
The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.
We welcome any comments about our site or worksheets on the Facebook comments box at the bottom of every page.
New! Comments
Have your say about the Math resources on this page! Leave me a comment in the box below.