GCSE Maths Geometry and Measure

Circles, Sectors and Arcs

Perimeter of a Sector

# Perimeter of a Sector

Here we will learn about calculating the perimeter of a sector including how to find the perimeter of a sector with a given angle and radius, and how to find the radius of a circle when given the perimeter of the sector.

There are also perimeter of a sector worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is the perimeter of a sector?

The perimeter of a sector is the distance around a sector.

We can calculate the perimeter of a sector by adding together the lengths of the two radii and the arc length of the sector.

E.g.

Calculate the perimeter of the below sector to 1 decimal place:

Arc length formula:

\text{Arc length} = \frac{\theta}{360} \times \pi\times d

\theta = Angle of the sector

r = Radius of the circle

Or

\text{Arc length} = \frac{\theta}{360} \times 2\times\pi \times r

\theta = Angle of the sector

r = Radius of the circle

Here angle \theta = 115^o and r = 8 , so

\begin{aligned} \text { Arc length }&=\frac{115}{360} \times 2 \times \pi \times 8 \\\\ &=16.05702912 \ldots \end{aligned}

To calculate the perimeter of the sector we need to add the arc length to the lengths of the two radii:

Perimeter of a sector formula:

\begin{aligned} \text { Perimeter of a sector }&= \text{Arc length + radius + radius}\\\\ &=16.05702912 + 8 + 8\\\\ &=32.05702912\ldots\\\\ &=32.1 \;cm \; (1.d.p) \end{aligned}

## How to find the perimeter of a sector

In order to find the perimeter of a sector:

1. Find the length of the diameter/radius.
2. Find the size of the angle creating the arc of the sector.
3. Find the arc length of the sector.

## Perimeter of a sector examples

### Example 1: calculate the perimeter of a sector (quadrant)

Calculate the perimeter of a sector of the sector.

1. Find the length of the diameter/radius.

2Find the size of the angle creating the arc of the sector.

Angle = 90^o . Shown by the symbol of the right angle.

3Find the arc length of the sector.

\text{Arc length } = \frac{\theta}{360} \times 2\times\pi \times r

\begin{aligned} &=\frac{90}{360} \times 2\times\pi \times 6 \\\\ &=3\pi \end{aligned}

Arc length: 3\pi \; cm

\text{Total perimeter of sector } = 3\pi + 6 +6 \; cm

\text{Total perimeter of sector } = 3\pi + 12 \; cm

\text{Perimeter of sector} = 3\pi + 12 \; cm

\text{Arc length} = 21.424777..cm

\text{Arc length} = 21.425 \; cm

Remember the perimeter of a sector is a measure of distance and therefore the units are not squared.

### Example 2: calculate the perimeter of a sector (semi circle)

Calculate the perimeter of a sector of the semicircle shown below.

Diameter = 24 \; cm

Angle = 180^o . This is because the shape shown is a semi circle. Therefore, the angle of the straight line is 180 degrees.

\text{Arc length } = \frac{\theta}{360} \times \pi\times d

\begin{aligned} &=\frac{180}{360} \times \pi\times 24 \\\\ &=12\pi \end{aligned}

Arc Length: 12\pi \; cm

Diameter: 24 \; cm

\text{Total perimeter of sector } = 12\pi + 12 +12 \; cm

\text{Total perimeter of sector } = 12\pi + 24 \; cm

\text{Perimeter of sector } = 12\pi + 24 \; cm

This answer is in terms of pi

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

Calculate the perimeter of a sector of the sector shown below.

Angle = 117^o

\text{Arc length } = \frac{\theta}{360} \times 2\times\pi \times r

\begin{aligned} &=\frac{117}{360} \times 2\times\pi \times 5.5 \\\\ &=\frac{143}{40} \pi \end{aligned}

Arc Length: \frac{143}{40} \pi \; cm

\text{Total perimeter of sector } = \frac{143}{40} \pi + 5.5 +5.5 \; cm

\text{Total perimeter of sector } = \frac{143}{40} \pi + 11 \; cm

\text{Perimeter of sector } = \frac{143}{40} \pi + 11 \; cm

\text{Perimeter of sector } = 22.2311...cm

\text{Perimeter of sector } = 22.2 \; cm

### Example 4: calculate the perimeter of a sector (the angle is not given)

Calculate the perimeter of a sector of the sector AOB below;

The length of the radius (OB) is 9cm.

The length of a chord (AB) is 10cm.

In this example you are not given the angle of the sector, you need to calculate it first.

Here you can use the triangle created by the two radii and the chord to find the angle (see below):

We will need to use the cosine rule to find the angle.

a^2=b^2+c^2-2bcCos(A)

A is the angle you are trying to find. You can therefore use the rearranged cosine rule to find the angle.

\begin{aligned} \operatorname{Cos} A&=\frac{b^{2}+c^{2}-a^{2}}{2 b c} \\ \operatorname{Cos} A&=\frac{19^{2}+19^{2}-20^{2}}{2 \times 19 \times 19} \\ \operatorname{Cos} A&=\frac{161}{361} \\ A&=\operatorname{Cos}^{-1}\left(\frac{161}{361}\right) \\ A&=63.51^{\circ} \end{aligned}

The size of the angle creating the sector (made by the two radii) is 63.5^o .

As you know the radius you can use the formula which has ‘r’ as a variable.

\text{Arc length } = \frac{\theta}{360} \times 2\times\pi \times r

\begin{aligned} &=\frac{63.5}{360} \times 2\times\pi \times 9 \\\\ &=21.057... \end{aligned}

Arc Length: 21.057 \; cm           Notice how you round this to more than the final                                                       answers required. This is to avoid rounding errors.

\text{Total perimeter of sector } = 21.057 + 19 + 19 \; cm

\text{Total perimeter of sector } = 59.057 \; cm

\text{Arc length } = 59.057 \; cm

\text{Arc length } = 59.1 \; cm

## How to solve problems involving the perimeter of a sector

Sometimes you may be given the perimeter of a sector and asked to find a property of the circle e.g the radius.

In this case you need to ‘reverse’ the process.

1. Clearly state which of the properties you know and do not know.
2. Create an equation for the perimeter of the sector and substitute in the value you know.
3. Solve the equation to find the unknown property.

## Solving problems involving perimeter of a sector examples

### Example 5: finding the radius given the perimeter of a sector

The sector below has a perimeter of 20cm and an angle of 125^o.

Calculate the length of x.

Give you answer to 2 decimal places.

Angle of sector = \; 125^o

Perimeter of a sector = \; 20cm

Arc length = \; \frac{125}{360} \times 2\times\pi \times x

Arc length = \; \frac{25x}{36} \times\pi

\begin{aligned} 20&=\frac{25}{36} \pi x+x+x \hspace{2.6cm} \text{Substitute in values } \\\\ 20&=\frac{25}{36} \pi x+2 x \hspace{3cm} \text{Simplify } x+x \end{aligned}

\begin{aligned} 20&=\frac{25}{36} \pi x+2 x \hspace{3cm} \text{Simplify } x+x \\\\ 20&=x\left(\frac{25}{36} \pi+2\right) \hspace{2.5cm} \text{Take } x \text{ out as a factor} \\\\ 20&=x(4.18166) \\\\ 20 \div 4.18166&=x \hspace{4.3cm} \text{Make } x \text{ the subject}\\\\ 4.783 \ldots&=x \end{aligned}

x=4.783...

x=4.78

### Example 6: finding the angle of a sector given the perimeter of a sector

The sector below has an perimeter 62cm and a radius of 18.

Calculate the length of x.

Give you answer to 2 decimal places.

Angle of sector = \; x^o

Perimeter of a sector = \; 62cm

\begin{aligned} \text { Arc length }&=\frac{x}{360} \times 2 \times \pi \times 18 \\\\ \text { Arc length }&= \frac{1}{10} \pi x \end{aligned}

\begin{aligned} 62&=\frac{1}{10} \pi x+18+18 \hspace{2.3cm} \text{Substitute in values } \\\\ 62&=\frac{1}{10} \pi x+36 \hspace{3cm} \text{Simplify} \\\\ 26&=\frac{1}{10} \pi x \end{aligned}

\begin{aligned} 26&=\frac{1}{10} \pi x \\\\ 26 \div \frac{\pi}{10}&=x \hspace{4.8cm} \text{Make } x \text{ the subject}\\\\ 82.76057&=x \end{aligned}

x=82.76057...

x=82.76

### Common misconceptions

• Finding the length of the arc not the perimeter of the sector

You must remember that the perimeter of the sector is the combined length of the arc and the two radii.

• Units

Remember the perimeter is a length and therefore the units will not be squared or cubed. Square units are for area and cubed units are for volume.

Perimeter of a sector is part of our series of lessons to support revision on circles, sectors and arcs. You may find it helpful to start with the main circles, sectors and arcs lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

### Practice perimeter of a sector questions

1. What is the specific name for the perimeter of a sector with an angle of 360^o

Circumference

Sector

Arc

Minor arc

A sector with an angle of 360º is a full circle.

Therefore, the perimeter of the sector is the whole length of the outside of the circle which is called the circumference.

2. The perimeter of a sector is made up of…

The arc only

The perimeter of a shape is the distance around a two-dimensional shape.

Therefore, the perimeter of a sector is the combined length of the arc and two radii.

3. Calculate the perimeter of a sector of this sector, in terms of pi?

5\pi+20 \; cm

5\pi \; cm

25\pi \; cm^2

35.7 \; cm

Total perimeter of sector = Length of arc + radius + radius

Total perimeter of sector = 5\pi + 10 +10 \; cm

Total perimeter of sector = 5\pi + 20 \; cm

This answer is in terms of pi

4. Calculate the perimeter of a sector of this semi circle in terms of pi.

51.4 \; cm

10\pi \; cm

10\pi + 20 \; cm^2

10\pi + 20 \; cm

Total perimeter of sector = Length of arc + radius + radius

Total perimeter of sector = 10\pi + 10 +10 \; cm

Total perimeter of sector = 5\pi + 20 \; cm

This answer is in terms of pi

5. Calculate the perimeter of a sector of this sector in terms of pi.

107.4 \; cm

192\pi \; cm^2

24\pi +32 \; cm

24\pi \; cm

Total perimeter of sector = Length of arc + radius + radius

Total perimeter of sector = 24\pi + 16 +16 \; cm

Total perimeter of sector = 5\pi + 32 \; cm

This answer is in terms of pi

6. Calculate the perimeter of a sector of this sector to 1 decimal place.

30.0 \; cm

21.99 \; cm

14\pi \; cm^2

7\pi +8 \; cm

Total perimeter of sector = Length of arc + radius + radius

Total perimeter of sector = 7\pi + 4 +4 \; cm

Total perimeter of sector = 7\pi + 8 \; cm

Total perimeter of sector = 29.9911 \; cm

Total perimeter of sector = 30.0 \; cm

### Perimeter of a sector GCSE questions

1. The diagram shows a sector of a circle with centre O .

The radius of the circle is 13cm.

The angle of the sector is 150^o.

Calculate the perimeter of a sector, give your answer correct to 3 significant figures

(4 marks)

(1)

“34.0339” + 13 + 13

(1)

60.0339

(1)

60.0

(1)

2. The diagram shows a sector of a circle with centre O .

The radius of the circle is 6.5cm.

The acute angle AOB is 70^o

Calculate the perimeter of a sector, give your answer correct to 3 significant figures.

(4 marks)

(1)

“32.899” + 13 + 13

(1)

58.899

(1)

58.9

(1)

3. The diagram shows a sector of a circle with centre O ..

The radius of the circle is 4.9cm.

The acute angle AOB is 60^o

Calculate the perimeter of a sector, give your answer correct to 2 significant figures.

(4 marks)

(1)

“5.1312” + 4.9 + 4.9

(1)

14.9312

(1)

14.93

(1)

4. Below is a sector of a circle with an perimeter of a sector of 60cm and a radius of 10cm

Find the size of the angle labelled x.

Give you answer correct to 3 significant figures.

(5 marks)

Correct attempt to form an equation for the perimeter of a sector

60=\frac{x}{360} \times 2 \times \boldsymbol{\pi} \times 10 + 20

(1)

Attempt to solve equation for x

40 =\frac{x}{18} \pi

40 \div \frac{\pi}{18} = x

(2)

229.18

(1)

229

(1)

5. The sector below has a perimeter of a sector of 30cm.

Calculate the length of x.

(5 marks)

Correct attempt to form an equation for the perimeter of a sector

30=\frac{135}{360} \times 2 \times \boldsymbol{\pi} \times x +2x

(1)

Attempt to solve equation for x

\begin{aligned} &30=\frac{3}{4} \pi x + 2x \\\\ &30 =x(\frac{3}{4} \pi + 2x ) \\\\ &30 =x(4.356) \\\\ \end{aligned}

(2)

6.8867

(1)

6.89

(1)

## Learning checklist

You have now learned how to:

• Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference
• Identify and apply circle definitions and properties, including tangent, arc, sector and segment
• Calculate the length of an arc
• Calculate the perimeter of a sector
• Calculate a property of a circle given the perimeter of a sector

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