Sector Areas and Arc Lengths

What you'll learn

How a sector is linked to a fraction of a full circle.
How to calculate sector area and arc length.
How to find the perimeter of a sector.
How to handle shaded regions and questions where the arc length is given.

1. Circle facts you need first

A circle has a centre, and the radius is the distance from the centre to the edge. The diameter goes all the way across the circle through the centre, so it is twice the radius.

A circle showing the centre, one radius, and the diameter through the centre.

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

For a circle with radius $r$ :

area=πr2circumference=2πr\begin{aligned} \text{area} &= \pi r^2 \\ \text{circumference} &= 2\pi r \end{aligned}

Example

Full circle warm-up

A circle has radius 6 cm. For the circumference, substitute $r = 6$ :
$C=2π×6=12πC=2\pi \times 6=12\pi$
For the area, substitute $r = 6$ :
$A=π×62=36πA=\pi \times 6^2=36\pi$
So the circumference is $12π12\pi$ cm and the area is $36π36\pi$ cm².

2. What is a sector?

A sector is a slice of a circle, made from two radii and the curved edge between them.

A sector is formed by two radii and the arc between them, with the central angle at the centre.

Definition

Sector language

An arc is part of the circumference of a circle.
The central angle is the angle at the centre of the circle.
A minor sector is less than half a circle.
A major sector is more than half a circle.

A full turn is 360°. So a sector with angle $θ\theta$ is this fraction of the full circle:

θ360\frac{\theta}{360}

Key Idea

The sector fraction

The same fraction, $θ360\frac{\theta}{360}$ , is used for both sector area and arc length.

Example

Finding the fraction of a circle

A sector has angle 120°. Write the angle over 360:

A 120° sector is one third of a full circle.

$$
\frac{120}{360}
$$

2. Simplify the fraction:

$$
\frac{120}{360}=\frac{1}{3}
$$

3. So a 120° sector is one third of the full circle.

3. Area of a sector

To find the area of a sector, take the same fraction of the full circle area.

area=θ360×πr2\text{sector area}=\frac{\theta}{360}\times \pi r^2

Here, $r$ is the radius and $θ\theta$ is the angle at the centre.

Example

Area of a sector to 3 significant figures

A sector has radius 8 cm and angle 135°. Use the formula:

Sector with radius 8 cm and central angle 135° for finding the sector area.

$$
A=\frac{135}{360}\times \pi \times 8^2
$$

2. Simplify and calculate:

$$
A=24\pi \approx 75.398
$$

3. Round to 3 significant figures.

The area is 75.4 cm².

Common Mistake

Using the wrong angle

If the sector angle is 220°, it is a major sector. Use $220360\frac{220}{360}$ unless the question asks for the smaller remaining part.

4. Arc length

The arc length is the length of the curved edge of the sector.

To find it, take the same fraction of the full circumference.

length=θ360×2πr\text{arc length}=\frac{\theta}{360}\times 2\pi r

Example

Arc length in terms of π

A sector has radius 10 cm and angle 108°. Substitute into the arc length formula:

Sector with radius 10 cm and central angle 108°, with the curved arc length to find.

$$
L=\frac{108}{360}\times 2\pi \times 10
$$

2. Simplify:

$$
L=\frac{3}{10}\times 20\pi=6\pi
$$

3. The arc length is $6π6\pi$ cm.

Tip

In terms of π

If the question asks for an answer “in terms of π”, leave π in your answer instead of using a decimal.

5. Perimeter of a sector

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

For a sector, the outside is made from:

one arc
two radii

So:

length+2r\text{perimeter}=\text{arc length}+2r

Example

Perimeter of a sector

A sector has radius 4.8 cm and angle 75°. First find the arc length:

The perimeter of a sector is the arc plus the two straight radii.

$$
L=\frac{75}{360}\times 2\pi \times 4.8=2\pi
$$

2. Add the two radii:

$$
P=2\pi+4.8+4.8
$$

3. Calculate the decimal value:

$$
P\approx 15.883
$$

4. The perimeter is 15.9 cm to 3 significant figures.

Common Mistake

Only finding the arc

For perimeter questions, do not stop after finding the arc length. You must add both straight sides as well.

6. Shaded regions with semicircles

A semicircle is half a circle. If you are given its diameter, halve it to get the radius.

For shaded area questions, use:

area\text{shaded area}=\text{total area}-\text{unshaded area}

Example

Percentage of a rectangle shaded

A rectangle is 14 cm by 10 cm. Its area is:

A 14 cm by 10 cm rectangle containing an unshaded semicircle of diameter 10 cm, leaving the rest shaded.

$$
14\times 10=140
$$

2. A semicircle inside it has diameter 10 cm, so its radius is 5 cm. Find the semicircle area:

$$
\frac{1}{2}\times \pi \times 5^2=12.5\pi
$$

3. Subtract to find the shaded area:

$$
140-12.5\pi \approx 100.730
$$

4. Find the percentage shaded:

$$
\frac{100.730}{140}\times 100\approx 71.9
$$

5. The shaded area is 71.9% of the rectangle, to 1 decimal place.

7. Finding sector area when arc length is given

Sometimes you are given the radius and the arc length, but not the angle.

For a sector:

A=12rLA=\frac{1}{2}rL

where $L$ is the arc length.

Example

Area from arc length

A sector has radius 8 cm and arc length $5π5\pi$ cm. Use the shortcut formula:

When the arc length is given, the sector area can be found directly from the radius and arc length.

$$
A=\frac{1}{2}rL
$$

2. Substitute $r = 8$ and $L=5πL=5\pi$ :

$$
A=\frac{1}{2}\times 8\times 5\pi
$$

3. Simplify:

$$
A=20\pi
$$

4. The area of the sector is $20π20\pi$ cm².

Exam technique

In the exam

Decide whether the question wants area, arc length, perimeter, or shaded area before choosing a formula.
For sector area and arc length, always start with the fraction $θ360\frac{\theta}{360}$ .
Check the requested form: exact in terms of π, 3 significant figures, or 1 decimal place.

Self review

Check yourself

If a sector has angle 90°, what fraction of the full circle is it?
What three lengths make up the perimeter of a sector?
If a semicircle has diameter 18 cm, what radius should you use in the area formula?

Sector Areas and Arc Lengths

What you'll learn

How a sector is linked to a fraction of a full circle.
How to calculate sector area and arc length.
How to find the perimeter of a sector.
How to handle shaded regions and questions where the arc length is given.

1. Circle facts you need first

A circle has a centre, and the radius is the distance from the centre to the edge. The diameter goes all the way across the circle through the centre, so it is twice the radius.

A circle showing the centre, one radius, and the diameter through the centre.

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

For a circle with radius $r$ :

area=πr2circumference=2πr\begin{aligned} \text{area} &= \pi r^2 \\ \text{circumference} &= 2\pi r \end{aligned}

Example

Full circle warm-up

A circle has radius 6 cm. For the circumference, substitute $r = 6$ :
$C=2π×6=12πC=2\pi \times 6=12\pi$
For the area, substitute $r = 6$ :
$A=π×62=36πA=\pi \times 6^2=36\pi$
So the circumference is $12π12\pi$ cm and the area is $36π36\pi$ cm².

2. What is a sector?

A sector is a slice of a circle, made from two radii and the curved edge between them.

A sector is formed by two radii and the arc between them, with the central angle at the centre.

Definition

Sector language

An arc is part of the circumference of a circle.
The central angle is the angle at the centre of the circle.
A minor sector is less than half a circle.
A major sector is more than half a circle.

A full turn is 360°. So a sector with angle $θ\theta$ is this fraction of the full circle:

θ360\frac{\theta}{360}

Key Idea

The sector fraction

The same fraction, $θ360\frac{\theta}{360}$ , is used for both sector area and arc length.

Example

Finding the fraction of a circle

A sector has angle 120°. Write the angle over 360:

A 120° sector is one third of a full circle.

$$
\frac{120}{360}
$$

2. Simplify the fraction:

$$
\frac{120}{360}=\frac{1}{3}
$$

3. So a 120° sector is one third of the full circle.

3. Area of a sector

To find the area of a sector, take the same fraction of the full circle area.

area=θ360×πr2\text{sector area}=\frac{\theta}{360}\times \pi r^2

Here, $r$ is the radius and $θ\theta$ is the angle at the centre.

Example

Area of a sector to 3 significant figures

A sector has radius 8 cm and angle 135°. Use the formula:

Sector with radius 8 cm and central angle 135° for finding the sector area.

$$
A=\frac{135}{360}\times \pi \times 8^2
$$

2. Simplify and calculate:

$$
A=24\pi \approx 75.398
$$

3. Round to 3 significant figures.

The area is 75.4 cm².

Common Mistake

Using the wrong angle

If the sector angle is 220°, it is a major sector. Use $220360\frac{220}{360}$ unless the question asks for the smaller remaining part.

4. Arc length

The arc length is the length of the curved edge of the sector.

To find it, take the same fraction of the full circumference.

length=θ360×2πr\text{arc length}=\frac{\theta}{360}\times 2\pi r

Example

Arc length in terms of π

A sector has radius 10 cm and angle 108°. Substitute into the arc length formula:

Sector with radius 10 cm and central angle 108°, with the curved arc length to find.

$$
L=\frac{108}{360}\times 2\pi \times 10
$$

2. Simplify:

$$
L=\frac{3}{10}\times 20\pi=6\pi
$$

3. The arc length is $6π6\pi$ cm.

Tip

In terms of π

If the question asks for an answer “in terms of π”, leave π in your answer instead of using a decimal.

5. Perimeter of a sector

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

For a sector, the outside is made from:

one arc
two radii

So:

length+2r\text{perimeter}=\text{arc length}+2r

Example

Perimeter of a sector

A sector has radius 4.8 cm and angle 75°. First find the arc length:

The perimeter of a sector is the arc plus the two straight radii.

$$
L=\frac{75}{360}\times 2\pi \times 4.8=2\pi
$$

2. Add the two radii:

$$
P=2\pi+4.8+4.8
$$

3. Calculate the decimal value:

$$
P\approx 15.883
$$

4. The perimeter is 15.9 cm to 3 significant figures.

Common Mistake

Only finding the arc

For perimeter questions, do not stop after finding the arc length. You must add both straight sides as well.

6. Shaded regions with semicircles

A semicircle is half a circle. If you are given its diameter, halve it to get the radius.

For shaded area questions, use:

area\text{shaded area}=\text{total area}-\text{unshaded area}

Example

Percentage of a rectangle shaded

A rectangle is 14 cm by 10 cm. Its area is:

A 14 cm by 10 cm rectangle containing an unshaded semicircle of diameter 10 cm, leaving the rest shaded.

$$
14\times 10=140
$$

2. A semicircle inside it has diameter 10 cm, so its radius is 5 cm. Find the semicircle area:

$$
\frac{1}{2}\times \pi \times 5^2=12.5\pi
$$

3. Subtract to find the shaded area:

$$
140-12.5\pi \approx 100.730
$$

4. Find the percentage shaded:

$$
\frac{100.730}{140}\times 100\approx 71.9
$$

5. The shaded area is 71.9% of the rectangle, to 1 decimal place.

7. Finding sector area when arc length is given

Sometimes you are given the radius and the arc length, but not the angle.

For a sector:

A=12rLA=\frac{1}{2}rL

where $L$ is the arc length.

Example

Area from arc length

A sector has radius 8 cm and arc length $5π5\pi$ cm. Use the shortcut formula:

When the arc length is given, the sector area can be found directly from the radius and arc length.

$$
A=\frac{1}{2}rL
$$

2. Substitute $r = 8$ and $L=5πL=5\pi$ :

$$
A=\frac{1}{2}\times 8\times 5\pi
$$

3. Simplify:

$$
A=20\pi
$$

4. The area of the sector is $20π20\pi$ cm².

Exam technique

In the exam

Decide whether the question wants area, arc length, perimeter, or shaded area before choosing a formula.
For sector area and arc length, always start with the fraction $θ360\frac{\theta}{360}$ .
Check the requested form: exact in terms of π, 3 significant figures, or 1 decimal place.

Self review

Check yourself

If a sector has angle 90°, what fraction of the full circle is it?
What three lengths make up the perimeter of a sector?
If a semicircle has diameter 18 cm, what radius should you use in the area formula?

Sector Areas and Arc Lengths

What you'll learn

1. Circle facts you need first

2. What is a sector?

3. Area of a sector

4. Arc length

5. Perimeter of a sector

6. Shaded regions with semicircles

7. Finding sector area when arc length is given

Recap questions

You've reached the end

Sector Areas and Arc Lengths

What you'll learn

1. Circle facts you need first

2. What is a sector?

3. Area of a sector

4. Arc length

5. Perimeter of a sector

6. Shaded regions with semicircles

7. Finding sector area when arc length is given

Recap questions

You've reached the end

Sector Areas and Arc Lengths

What you'll learn

1. Circle facts you need first

2. What is a sector?

3. Area of a sector

4. Arc length

5. Perimeter of a sector

6. Shaded regions with semicircles

7. Finding sector area when arc length is given

Recap questions

You've reached the end

Sector Areas and Arc Lengths Revision Guide

Sector Areas and Arc Lengths

What you'll learn

1. Circle facts you need first

2. What is a sector?

3. Area of a sector

4. Arc length

5. Perimeter of a sector

6. Shaded regions with semicircles

7. Finding sector area when arc length is given

Recap questions

You've reached the end

Sector Areas and Arc Lengths Revision Guide