Area of a Sector Calculator
Welcome to our Area of a Sector Calculator page.
We explain how to find the area 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 ...
Area of a Sector Calculator
This calculator finds the area of a sector when the radius (or diameter) and the angle are known.
How the 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.
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 the area is slightly different.
Area of a Sector Formula
The area of a sector of a circle where the central angle is in degrees is: \[ A = {\theta \over 360} \pi r^2 \]
The area of a sector of a circle where the central angle is in radians is: \[ A = {\theta \over 2} r^2 \]
where θ is the central angle and r is the radius of the circle.
Area of a Sector Examples
Area of a Sector Example 1
Find the area of the sector below to 1 decimal place.
The sector has a central angle of 115° and a radius of 6 cm.
The area of a sector:
\[ A = {\theta \over 360} \pi r^2 \] where A is the area, θ is the central angle and r is the radius of the circle.
So if we substitute the values of the angle and radius into this equation, we get: \[ A = {115 \over 360} \pi \cdot 6^2 = {115 \over 360} \pi \cdot 36 = {23 \over 2} \pi \]
This gives us a final answer of: \[ A = 36.1 \; cm^2 \; to \; 1 \; decimal \; place \]
Area of a Sector Example 2
Find the area of the sector below to 1 decimal place.
The sector has a central angle of 72° and a radius of 8 ½ in.
The area of a sector:
\[ A = {\theta \over 360} \pi r^2 \] where A is the area, θ is the central angle and r is the radius of the circle.
So if we substitute the values of the angle and radius into this equation, we get: \[ A = {72 \over 360} \pi \cdot (8 {1 \over 2})^2 = {72 \over 360} \pi \cdot {289 \over 4} = {289 \over 20} \pi \]
This gives us a final answer of: \[ A = 45.4 \; in^2 \; to \; 1 \; decimal \; place \]
Area of a Sector Example 3
Find the area 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 area of a sector:
\[ A = {\theta \over 360} \pi r^2 \] where A is the area, θ is the central angle and r is the radius of the circle.
So if we substitute the values of the angle and radius into this equation, we get: \[ A = {278 \over 360} \pi \cdot (9.6)^2 = {278 \over 360} \pi \cdot 92.16 = 71.168 \pi \]
This gives us a final answer of: \[ A = 223.6 \; cm^2 \; to \; 1 \; decimal \; place \]
Area of a Sector Example 4
Find the area 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 area of a sector, where the angle is in radians, is:
\[ A = {\theta \over 2} r^2 \] where A is the area, θ is the central angle and r is the radius of the circle.
So if we substitute the values of the angle and radius into this equation, we get: \[ A = {2 {1 \over 2} \over 2} \cdot 16^2 = {2 {1 \over 2} \over 2} \cdot 256 = 320 \]
This gives us a final answer of: \[ A = 320 \; m^2 \]
Area of a Sector Worksheets
We have created some worksheets to help you practice this skill.
The first and second sheets involve working out the area of different sectors where the angle is measured in degrees.
Sheet 3 involves finding the area of different sectors where the angle is measured in radians.
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