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# Sector of a Circle

A sector of a circle is an area of a circle where two of the sides are radii. An example of the sector (in red) is shown below:

A sector of a circle -StudySmarter Originals

An arc length is a part of the circle's circumference (perimeter). For the same sector, we could have arc as shown in green:

Arc length of a circle - StudySmarter Originals

## Circle sector theorems where the angle is in degrees

You might already be familiar with this but let's look at calculating the area and arc length of a circle sector when the angle is given in degrees.

### Calculating the area of a sector of a circle

The formula to calculate the area of a sector with an angle is:

where r is the radius of the circle

Circle A has a diameter of 10cm. A sector of circle A an angle of 50. What is the area of this sector?

• First, we need to calculate the radius of the circle. This is because the formula for the area of a sector uses this value rather than the diameter.

=

• Then, substitute your values into the area of a sector formula.

### Calculating the arc length of a sector of a circle

The formula to calculate the arc length of a sector with an angle is:

where d is the diameter of the circle:

Circle B has a radius of 12cm. A sector within Circle B has an angle of 100. What is the length of the arc length of this sector?

• First, the formula for the arc length of a sector requires the diameter of the circle rather than the radius.
• Then, you can substitute your values from the question into the formula

## Circle sector theorems where the angle is in radians

• You also need to be able to calculate the arc length and area of a sector of a circle where the angle is given in radians.

• Radians are an alternative unit to degrees that we can use to measure an angle at the centre of the circle.

• To recap, some common degree to radian conversions.

### Calculating the area of a sector of a circle

To calculate the area of a sector of a circle with an angle , the formula you use is:

where r is the radius of the circle.

Circle C has a radius of 15cm. Within Circle C, there is a sector with an angle of 0.5 radians. What is the area of this sector?

• As all the variables are in the form required in the formula, you can substitute their values into the formula.

### Calculating the arc length of a sector of a circle

To calculate the arc length of a sector of a circle with an angle ,

the formula you use is:

where r is the radius of the circle.

A sector in Circle D has an angle of 1.2 radians. Circle D has a diameter of 19. What is the arc length of this sector?

• The formula requires the radius rather than the diameter.
• You can then substitute these values into the formula

## Sector of a Circle - Key takeaways

• A sector of a circle is the proportion of a circle where two of the sides are radii. An arc length of the sector is the proportion of the circumference which runs the length of the sector of the circle.
• If the angle at the centre of the circle is in degrees, the formula for finding the area of the sector is: . To calculate the arc length, the formula is:
• If the angle of the circle is in radians, the formula for finding the area of the sector is: . For calculating the arc length of the sector, the formula is

A sector of a circle is a proportion of a circle where two sides are radii.

To find the sector of a circle you need to use one of the formulas for the area of the sector. Which one you use is dependent on whether the angle at the centre is in radians or in degrees.

There are two formulas of a sector. One is to calculate the area of a sector of a circle. Area of a sector= pi × r^2 × (θ /360). The other one is to find the arc length of the sector of the circle. Arc length = pi × d × (θ /360)

## Final Sector of a Circle Quiz

Question

What is a sector?

A sector of a circle is a proportion of a circle where two slides are radii

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Question

How do you find the sector of a circle?

To find the sector of a circle you need to use one of the formulas for the area of the sector. Which one you use is dependent on whether the angle at the centre is in radians or in degrees

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Question

There are two formulas to find the area of a sector of a circle. What decides which one you use?

Whether the angle at the centre of the circle is in degrees or radians

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Question

What is an arc length?

A proportion of the circle’s circumference, which is defined by the sector.

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Question

Circle A has a diameter of 10cm. A sector of circle A has an angle of 120. What is the area of this sector?

105 (3 s.f)

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Question

Circle B has a diameter of 4cm. A sector of circle B has an angle of 20. What is the area of this sector?

2.79 (3 s.f)

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Question

Circle D has a radius of 10cm. A sector of circle D has an angle of 60. What is the arc length of this sector?

10.5 (3 s.f.).

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Question

Circle E has a diameter of 15cm. A sector of circle E has an angle of 90. What is the arc length of this sector?

11.8 (3 s.f.)

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Question

An alternative to degrees for measuring angles.

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Question

Circle F has a diameter of 9cm. A sector of circle E has an angle of 115. What is the arc length of this sector?

9.03 (3 s.f.)

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Question

Circle G has a radius of 5cm. Within Circle G, there is a sector with an angle of 1.2 radians. What is the area of this sector?

15

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Question

Circle H has a radius of 11cm. Within Circle H, there is a sector with the angle x radians and an area of 222.9cm2. What is the value for the angle of this sector?

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