Sector Areas and Arc Lengths

What you'll learn

What a sector is: a slice of a circle made from two radii and an arc.
How to find the area of a sector using the angle at the centre.
How to find an arc length, and then the perimeter of a sector.
How to handle shaded regions involving sectors, rectangles and semicircles.

The circle facts you need first

A circle is the set of points the same distance from a fixed middle point, called the centre. The distance from the centre to the edge is the radius. The distance all the way across through the centre is the diameter, which is twice the radius.

The circumference is the distance around the outside of a full circle. The area is the space inside the circle.

For a full circle:

Circumference: $2\pi r$
Area: $\pi r^2$

where $r$ is the radius.

Example

Recapping full circle formulae

Suppose a circle has radius 6 cm.
The circumference is found using $2\pi r$ :
$2\pi \times 6 = 12\pi$
The area is found using $\pi r^2$ :
$\pi \times 6^2 = 36\pi$
So the circumference is $12π12\pi$ cm and the area is $36π36\pi$ cm².

What is a sector?

A sector is part of a circle enclosed by two radii and an arc. An arc is part of the circumference. The angle between the two radii at the centre is called the central angle.

This diagram shows the key parts you need to recognise.

Labelled sector diagram showing centre, radius, central angle, arc length and shaded sector area

Definition

Sector vocabulary

A radius is a straight line from the centre to the circumference.
An arc is the curved edge of a sector.
A sector is a fraction of a circle.
The central angle is the angle at the centre; we often call it $θ\theta$ .

The key idea: use the fraction of the circle

A full circle is 360°. So a sector with angle $θ\theta$ is the fraction $θ360\frac{\theta}{360}$ of the whole circle.

Key Idea

Sector fraction

For any sector:

circle=θ360\text{fraction of circle} = \frac{\theta}{360}

Use this same fraction for both area and arc length.

Example

Finding the fraction of a circle

A sector has angle 135°.
Write the sector as a fraction of a full circle:
$135360\frac{135}{360}$
Simplify the fraction:
$135360=38\frac{135}{360} = \frac{3}{8}$
So the sector is $38\frac{3}{8}$ of the full circle.

Area of a sector

To find the area of a sector, take the fraction of the circle and multiply it by the area of the full circle.

\frac{\theta}{360}\times \pi r^2

where $A$ is the sector area, $r$ is the radius, and $θ\theta$ is the central angle in degrees.

Example

Area of a sector to 3 significant figures

A sector has radius 8 cm and angle 150°.
Use the sector area formula:
$\frac{150}{360}\times \pi \times 8^2$
Calculate the exact expression first:
$\frac{150}{360}\times 64\pi = \frac{80\pi}{3}$
Convert to a decimal:
$80π3≈83.7758\frac{80\pi}{3} \approx 83.7758$
Round to 3 significant figures: 83.8 cm².

Common Mistake

Rounding too early

Keep the full calculator value until the final answer. If you round halfway through, your 3 significant figure answer may be slightly wrong.

Larger sectors

A major sector is a sector bigger than a semicircle, so its angle is greater than 180°. The same formula still works: just use the angle you are given.

For example, a 200° sector is more than half a circle, because 200° is more than 180°.

Arc length

The arc length is the length of the curved part of the sector. Since an arc is a fraction of the circumference, use the same fraction idea:

\frac{\theta}{360}\times 2\pi r

where $L$ is the arc length.

Example

Arc length in terms of pi

A sector has radius 12 cm and angle 135°.
Use the arc length formula:
$\frac{135}{360}\times 2\pi \times 12$
Simplify the fraction:
$\frac{3}{8}\times 24\pi$
Multiply:
$9\pi$
So the arc length is $9π9\pi$ cm.

Tip

When the question says in terms of pi

Do not press the calculator button for $π\pi$ at the end. Leave your answer with $π\pi$ in it, such as $9π9\pi$ cm or $25π2\frac{25\pi}{2}$ cm.

Perimeter of a sector

The perimeter of a shape is the total distance around its outside.

For a sector, the outside is made from:

one arc
two radii

So:

sector=L+2r\text{perimeter of sector} = L + 2r

Example

Finding the perimeter of a sector

A sector has radius 5.4 cm and angle 80°.
First find the arc length:
$\frac{80}{360}\times 2\pi \times 5.4$
Simplify:
$2.4\pi$
Add the two radii as well as the arc:
$2.4\pi + 5.4 + 5.4$
Calculate and round to 3 significant figures:
$2.4\pi \approx 18.3398$
The perimeter is 18.3 cm.

Common Mistake

Forgetting the two straight sides

For perimeter, do not just find the arc length. A sector has two straight sides as well, so add both radii.

Shaded regions with sectors and semicircles

A semicircle is half a circle. Its area is half of $πr2\pi r^2$ .

Many IGCSE questions ask for a shaded region made from two shapes. The main strategy is:

Find the area of the larger shape.
Find the area of the smaller shape.
Subtract.

Here is the kind of “sector minus semicircle” structure you may see.

Composite shape diagram showing a sector and a semicircle with the shaded region found by subtraction

Example

Sector minus semicircle

A 90° sector has radius 10 cm. A semicircle is drawn with diameter 10 cm inside it.
Find the area of the sector:
$Asector=90360×π×102=25πA_{\text{sector}} = \frac{90}{360}\times \pi \times 10^2 = 25\pi$
The semicircle has diameter 10 cm, so its radius is 5 cm.
Find the area of the semicircle:
$Asemi=12×π×52=25π2A_{\text{semi}} = \frac{1}{2}\times \pi \times 5^2 = \frac{25\pi}{2}$
Subtract the semicircle from the sector:
$Ashaded=25π−25π2=25π2A_{\text{shaded}} = 25\pi - \frac{25\pi}{2} = \frac{25\pi}{2}$
The shaded area is $25π2\frac{25\pi}{2}$ cm².

Example

Percentage shaded in a rectangle

A rectangle is 16 cm by 10 cm. A semicircle with diameter 10 cm is cut out from it.
Find the area of the rectangle:
$16×10=16016\times 10 = 160$
The semicircle has radius 5 cm, so its area is:
$12×π×52=25π2\frac{1}{2}\times \pi \times 5^2 = \frac{25\pi}{2}$
Find the shaded area:
$\frac{25\pi}{2}$
Convert to a percentage of the rectangle:
$160−25π2160×100≈75.456\frac{160 - \frac{25\pi}{2}}{160}\times 100 \approx 75.456$
Rounded to 1 decimal place, the shaded percentage is 75.5%.

When you are given arc length instead of angle

Sometimes the angle is not given. If you know the radius and the arc length, you can find the sector area directly.

\frac{1}{2}rL

where $L$ is the arc length.

This works because the arc length and the sector area are both the same fraction of the full circle.

Example

Finding sector area from arc length

A sector has radius 8 cm and arc length $5π5\pi$ cm.
Use the formula $\frac{1}{2}rL$ .
Substitute $r = 8$ and $5\pi$ :
$\frac{1}{2}\times 8 \times 5\pi$
Simplify:
$20\pi$
So the sector area is $20π20\pi$ cm².

Exam technique

In the exam

Decide what you are finding first: area, arc length, perimeter, shaded area, or percentage.
Write the sector fraction $θ360\frac{\theta}{360}$ before substituting numbers.
If the answer says “in terms of $π\pi$ ”, leave $π\pi$ in your final answer; if it asks for significant figures, round only at the end.

Self review

Check yourself

Can you explain why a 90° sector is one quarter of a full circle?
When finding the perimeter of a sector, what three lengths must be included?
If you are given radius and arc length, which shortcut formula gives the sector area?

Sector Areas and Arc Lengths

What you'll learn

What a sector is: a slice of a circle made from two radii and an arc.
How to find the area of a sector using the angle at the centre.
How to find an arc length, and then the perimeter of a sector.
How to handle shaded regions involving sectors, rectangles and semicircles.

The circle facts you need first

The circumference is the distance around the outside of a full circle. The area is the space inside the circle.

For a full circle:

Circumference: $2\pi r$
Area: $\pi r^2$

where $r$ is the radius.

Example

Recapping full circle formulae

Suppose a circle has radius 6 cm.
The circumference is found using $2\pi r$ :
$2\pi \times 6 = 12\pi$
The area is found using $\pi r^2$ :
$\pi \times 6^2 = 36\pi$
So the circumference is $12π12\pi$ cm and the area is $36π36\pi$ cm².

What is a sector?

A sector is part of a circle enclosed by two radii and an arc. An arc is part of the circumference. The angle between the two radii at the centre is called the central angle.

This diagram shows the key parts you need to recognise.

Labelled sector diagram showing centre, radius, central angle, arc length and shaded sector area

Definition

Sector vocabulary

A radius is a straight line from the centre to the circumference.
An arc is the curved edge of a sector.
A sector is a fraction of a circle.
The central angle is the angle at the centre; we often call it $θ\theta$ .

The key idea: use the fraction of the circle

A full circle is 360°. So a sector with angle $θ\theta$ is the fraction $θ360\frac{\theta}{360}$ of the whole circle.

Key Idea

Sector fraction

For any sector:

circle=θ360\text{fraction of circle} = \frac{\theta}{360}

Use this same fraction for both area and arc length.

Example

Finding the fraction of a circle

A sector has angle 135°.
Write the sector as a fraction of a full circle:
$135360\frac{135}{360}$
Simplify the fraction:
$135360=38\frac{135}{360} = \frac{3}{8}$
So the sector is $38\frac{3}{8}$ of the full circle.

Area of a sector

To find the area of a sector, take the fraction of the circle and multiply it by the area of the full circle.

\frac{\theta}{360}\times \pi r^2

where $A$ is the sector area, $r$ is the radius, and $θ\theta$ is the central angle in degrees.

Example

Area of a sector to 3 significant figures

A sector has radius 8 cm and angle 150°.
Use the sector area formula:
$\frac{150}{360}\times \pi \times 8^2$
Calculate the exact expression first:
$\frac{150}{360}\times 64\pi = \frac{80\pi}{3}$
Convert to a decimal:
$80π3≈83.7758\frac{80\pi}{3} \approx 83.7758$
Round to 3 significant figures: 83.8 cm².

Common Mistake

Rounding too early

Keep the full calculator value until the final answer. If you round halfway through, your 3 significant figure answer may be slightly wrong.

Larger sectors

A major sector is a sector bigger than a semicircle, so its angle is greater than 180°. The same formula still works: just use the angle you are given.

For example, a 200° sector is more than half a circle, because 200° is more than 180°.

Arc length

The arc length is the length of the curved part of the sector. Since an arc is a fraction of the circumference, use the same fraction idea:

\frac{\theta}{360}\times 2\pi r

where $L$ is the arc length.

Example

Arc length in terms of pi

A sector has radius 12 cm and angle 135°.
Use the arc length formula:
$\frac{135}{360}\times 2\pi \times 12$
Simplify the fraction:
$\frac{3}{8}\times 24\pi$
Multiply:
$9\pi$
So the arc length is $9π9\pi$ cm.

Tip

When the question says in terms of pi

Do not press the calculator button for $π\pi$ at the end. Leave your answer with $π\pi$ in it, such as $9π9\pi$ cm or $25π2\frac{25\pi}{2}$ cm.

Perimeter of a sector

The perimeter of a shape is the total distance around its outside.

For a sector, the outside is made from:

one arc
two radii

So:

sector=L+2r\text{perimeter of sector} = L + 2r

Example

Finding the perimeter of a sector

A sector has radius 5.4 cm and angle 80°.
First find the arc length:
$\frac{80}{360}\times 2\pi \times 5.4$
Simplify:
$2.4\pi$
Add the two radii as well as the arc:
$2.4\pi + 5.4 + 5.4$
Calculate and round to 3 significant figures:
$2.4\pi \approx 18.3398$
The perimeter is 18.3 cm.

Common Mistake

Forgetting the two straight sides

For perimeter, do not just find the arc length. A sector has two straight sides as well, so add both radii.

Shaded regions with sectors and semicircles

A semicircle is half a circle. Its area is half of $πr2\pi r^2$ .

Many IGCSE questions ask for a shaded region made from two shapes. The main strategy is:

Find the area of the larger shape.
Find the area of the smaller shape.
Subtract.

Here is the kind of “sector minus semicircle” structure you may see.

Composite shape diagram showing a sector and a semicircle with the shaded region found by subtraction

Example

Sector minus semicircle

A 90° sector has radius 10 cm. A semicircle is drawn with diameter 10 cm inside it.
Find the area of the sector:
$Asector=90360×π×102=25πA_{\text{sector}} = \frac{90}{360}\times \pi \times 10^2 = 25\pi$
The semicircle has diameter 10 cm, so its radius is 5 cm.
Find the area of the semicircle:
$Asemi=12×π×52=25π2A_{\text{semi}} = \frac{1}{2}\times \pi \times 5^2 = \frac{25\pi}{2}$
Subtract the semicircle from the sector:
$Ashaded=25π−25π2=25π2A_{\text{shaded}} = 25\pi - \frac{25\pi}{2} = \frac{25\pi}{2}$
The shaded area is $25π2\frac{25\pi}{2}$ cm².

Example

Percentage shaded in a rectangle

A rectangle is 16 cm by 10 cm. A semicircle with diameter 10 cm is cut out from it.
Find the area of the rectangle:
$16×10=16016\times 10 = 160$
The semicircle has radius 5 cm, so its area is:
$12×π×52=25π2\frac{1}{2}\times \pi \times 5^2 = \frac{25\pi}{2}$
Find the shaded area:
$\frac{25\pi}{2}$
Convert to a percentage of the rectangle:
$160−25π2160×100≈75.456\frac{160 - \frac{25\pi}{2}}{160}\times 100 \approx 75.456$
Rounded to 1 decimal place, the shaded percentage is 75.5%.

When you are given arc length instead of angle

Sometimes the angle is not given. If you know the radius and the arc length, you can find the sector area directly.

\frac{1}{2}rL

where $L$ is the arc length.

This works because the arc length and the sector area are both the same fraction of the full circle.

Example

Finding sector area from arc length

A sector has radius 8 cm and arc length $5π5\pi$ cm.
Use the formula $\frac{1}{2}rL$ .
Substitute $r = 8$ and $5\pi$ :
$\frac{1}{2}\times 8 \times 5\pi$
Simplify:
$20\pi$
So the sector area is $20π20\pi$ cm².

Exam technique

In the exam

Decide what you are finding first: area, arc length, perimeter, shaded area, or percentage.
Write the sector fraction $θ360\frac{\theta}{360}$ before substituting numbers.
If the answer says “in terms of $π\pi$ ”, leave $π\pi$ in your final answer; if it asks for significant figures, round only at the end.

Self review

Check yourself

Can you explain why a 90° sector is one quarter of a full circle?
When finding the perimeter of a sector, what three lengths must be included?
If you are given radius and arc length, which shortcut formula gives the sector area?

Sector Areas and Arc Lengths

What you'll learn

The circle facts you need first

What is a sector?

The key idea: use the fraction of the circle

Area of a sector

Larger sectors

Arc length

Perimeter of a sector

Shaded regions with sectors and semicircles

When you are given arc length instead of angle

Recap questions

You've reached the end

Sector Areas and Arc Lengths

What you'll learn

The circle facts you need first

What is a sector?

The key idea: use the fraction of the circle

Area of a sector

Larger sectors

Arc length

Perimeter of a sector

Shaded regions with sectors and semicircles

When you are given arc length instead of angle

Recap questions

You've reached the end

Sector Areas and Arc Lengths

What you'll learn

The circle facts you need first

What is a sector?

The key idea: use the fraction of the circle

Area of a sector

Larger sectors

Arc length

Perimeter of a sector

Shaded regions with sectors and semicircles

When you are given arc length instead of angle

Recap questions

You've reached the end

Sector Areas and Arc Lengths Revision Guide

Sector Areas and Arc Lengths

What you'll learn

The circle facts you need first

What is a sector?

The key idea: use the fraction of the circle

Area of a sector

Larger sectors

Arc length

Perimeter of a sector

Shaded regions with sectors and semicircles

When you are given arc length instead of angle

Recap questions

You've reached the end

Sector Areas and Arc Lengths Revision Guide