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Area of a sector

Here you will learn how to calculate the area of a sector including how to identify a sector of a circle, form and use the formula for the area of a sector of a circle, and calculate the area of a sector in various scenarios.

Students will first learn about the area of a sector as part of geometry in high school.

What is the area of a sector?

The area of a sector is the space inside the section of the circle created by two radii and an arc. It is a fraction of the area of the entire circle.

Area of a Sector 1 US

Major sector: a major sector has a central angle which is more than katex is not defined

Minor sector: a minor sector has a central angle which is less than katex is not defined

[FREE] Common Core Practice Tests (Grades 3 to 6)

[FREE] Common Core Practice Tests (Grades 3 to 6)

[FREE] Common Core Practice Tests (Grades 3 to 6)

Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents. 40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

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[FREE] Common Core Practice Tests (Grades 3 to 6)

[FREE] Common Core Practice Tests (Grades 3 to 6)

[FREE] Common Core Practice Tests (Grades 3 to 6)

Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents. 40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

DOWNLOAD FREE

How to find the area of a sector

In order to find the area of a sector, you need to be able to find the area of a circle.

This is because the area of a sector is part of the total area of the circle.

‘How much’ of the circle is decided by the angle created by the two radii.

Area of a Sector 2 US

The sum of the angle around a point is equal to katex is not defined therefore the area of a sector is the fraction of the full circle’s area.

The angle is out of katex is not defined or katex is not defined where katex is not defined (theta) represents the angle, so you can multiply this by the area of the circle to calculate the area of a sector.

To calculate the area of a sector, use the following formula:

katex is not defined

katex is not defined – angle of the sector

katex is not defined – radius of the circle

What is the area of a sector?

What is the area of a sector?

Common Core State Standards

How does this relate to high school math?

  • High School – Geometry – Circles (HS.G.C.B.5)
    Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

How to find the area of a sector

To find the area of a sector:

  1. Determine the length of the radius katex is not defined
  2. Find the size of the angle creating the sector katex is not defined
  3. Substitute known values into the formula for the area of a sector and solve.

Area of a sector examples

Example 1: calculate the area of a sector (quadrant)

Calculate the area of the segment shown below.

Give your answer to the nearest thousandth.

Area of a Sector 3 US

  1. Determine the length of the radius katex is not defined

Radius: katex is not defined

2Find the size of the angle creating the sector katex is not defined

Angle: katex is not defined (shown by the symbol of the right angle).

3Substitute known values into the formula for the area of a sector and solve.

katex is not defined

Example 2: calculate the area of a sector (semicircle)

Calculate the area of the semicircle shown below.

Give your answer in terms of katex is not defined

Area of a Sector 4 US

Determine the length of the radius katex is not defined

Show step

Find the size of the angle creating the sector katex is not defined

Show step

Substitute known values into the formula for the area of a sector and solve.

Show step

Example 3: calculate the area of a sector (with an angle given)

Calculate the area of the sector shown below.

Give your answer to the nearest tenth.

Area of a Sector 5 US

Determine the length of the radius katex is not defined

Show step

Find the size of the angle creating the sector katex is not defined

Show step

Substitute known values into the formula for the area of a sector and solve.

Show step

Example 4: calculate the area of a sector (the angle is not given – cosine rule)

katex is not defined is a sector of a circle.

The length of the radius katex is not defined is katex is not defined

The length of a chord katex is not defined is katex is not defined

Calculate the area of the sector katex is not defined below.

Give your answer to the nearest hundredth.

Area of a Sector 6 US

Determine the length of the radius katex is not defined

Show step

Find the size of the angle creating the sector katex is not defined

Show step

Substitute known values into the formula for the area of a sector and solve.

Show step

How to solve problems involving an area of a sector

In order to determine missing angle or radius given the area of a sector:

  1. Clearly state which of the properties you know (radius/angle of sector, area of sector).
  2. Substitute known values into the formula for the area of a sector.
  3. Solve for the unknown value.

Example 5: finding the radius given the area of the sector

The sector below has an area of katex is not defined

Calculate the length of katex is not defined

Give your answer to the nearest hundredth.

Area of a Sector 8 US

Clearly state which of the properties you know (radius/angle of sector, area of sector).

Show step

Substitute known values into the formula for the area of a sector.

Show step

Solve for the unknown value.

Show step

Example 6: finding the angle of a sector given the area of the sector

The sector below has an area of katex is not defined

Calculate the length of katex is not defined

Give your answer to the nearest hundredth.

Area of a Sector 9 US

Clearly state which of the properties you know (radius/angle of sector, area of sector).

Show step

Substitute known values into the formula for the area of a sector.

Show step

Teaching tips for area of a sector

  • Provide worksheets with different circle and sector problems. Use activities that require cutting out sectors from paper circles.

  • For advanced students, connect this topic to trigonometry, especially when dealing with radian measures.

  • Ensure students understand the parts of a circle, including the center of the circle and the diameter and radius of a circle. Students should also know circle formulas for finding the area and circumference of the circle.

Easy mistakes to make

  • Confusing the segment/sector
    A segment is made from a chord while a sector will have lines (radii) coming from the origin.
    Sectors are a portion of a circle; it can be helpful to think of a sector as a pizza slice.

  • Finding the area of the whole circle not the area of the sector
    Remember to find the fraction of the circle that makes the sector, not the area of the whole circle. To find the area of a sector, use katex is not defined not katex is not defined.

  • Incorrect use of the cosine rule
    Many mistakes are made when applying other rules within a sector question, such as the cosine rule. Take your time with these parts and regularly ask if your answer makes sense within the context of the question. If you need to review the cosine rule, do so before continuing.

  • Using incorrect units
    The area of a sector, like any area, is measured in square units. The specific units depend on the units used for the radius of the circle. (For example, if the radius is measured in centimeters, the area would be measured in square centimeters.)

Practice area of a sector questions

1. Calculate the area of this sector in terms of katex is not defined

 

Area of a Sector 10 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True
katex is not defined

2. Calculate the area of this semicircle in terms of katex is not defined

 

Area of a Sector 11 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

Since this is a semicircle, you know the angle measure is katex is not defined

 

katex is not defined

3. Calculate the area of the sector. Round your answer to the nearest tenth.

 

Area of a Sector 12 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False
katex is not defined

 

katex is not defined

4. Calculate the area of this sector. Write your answer to the nearest hundredth.

 

Area of a Sector 13 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False
katex is not defined

5. The area of the sector is katex is not defined and the angle katex is not defined

 

Determine the radius of the sector to the nearest hundredth.

 

Area of a Sector 13 US

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False
katex is not defined

6. katex is not defined is a sector of a circle where katex is not defined and katex is not defined are two ends of the arc of the sector, and katex is not defined is the center. The chord katex is not defined and the radius katex is not defined Calculate the area of the sector to katex is not defined significant figures.

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

Use the cosine rule to determine the size of angle katex is not defined

 

katex is not defined rearranges to become katex is not defined

 

Here, katex is not defined

 

katex is not defined

 

Substitute the values of the radius katex is not defined and the angle katex is not defined into the formula for the area of a sector:

 

katex is not defined

Area of a sector FAQs

What is the area of the sector of a circle?

The area of a sector is the space inside the section of the circle created by two radii and an arc. It is a fraction of the area of the entire circle.

How do you find the area of a sector?

To find the area of a sector, you can use the area formula for a sector of a circle. Area of a sector formula katex is not defined

What is the difference between the area of a sector and the perimeter of a sector?

The area of a sector measures the space inside the sector, while the perimeter of a sector measures the distance around the sector, including the arc and the two radii.

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