Circumference of a circle

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In order to access this I need to be confident with:

Parts of a circle Irrational numbers Rounding decimals places Rounding significant figures

This topic is relevant for:

GCSE Maths Geometry and Measure 2D Shapes Circles, Sectors & Arcs Parts Of A Circle

Circumference Of A Circle

Here we will learn about calculating the circumference of a circle including how to calculate the circumference of a circle given the radius, diameter or area, how to calculate the perimeter of a semi-circle and how to calculate the radius or diameter of a circle given the circumference.

There are also circumference of a circle 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 circumference of a circle?

The circumference of a circle is the distance around the circle.

Circumference is a special case of perimeter. Both describe the total length of the boundary of a two dimensional shape, but circumference specifically refers to the perimeter of a curved figure or arc. Therefore it only applies to circles, ovals, ellipses, arcs, etc.

Circumference of a circle formula:
There is a specific relationship between the diameter of a circle and its circumference.
If you multiply the diameter of a circle by $\pi$ you will calculate the length of the circumference. This is true of all circles.

Therefore the below formula is for the circumference of any given circle:

\pi \times d

As the diameter of the circle is twice the radius we can also use the below formula:

2\times\pi \times r

Which formula you use will depend on whether you know the diameter or the radius of the circle.

E.g.
What is the circumference of a circle with radius $3cm$ ?

\begin{aligned} \text { Circumference } &=2 \times \pi \times r \\\\ &=2 \times \pi \times 3 \\\\ &=6 \pi \mathrm{cm} \\\\ &=18.8 \mathrm{~cm} \quad(1 . \mathrm{d} . \mathrm{p}) \end{aligned}

\begin{aligned} \text { Circumference } &=\pi \times d \\\\ &=\pi \times 6 \\\\ &=6 \pi \mathrm{cm} \\\\ &=18.8 \mathrm{~cm} \quad(1 . \mathrm{d} . \mathrm{p}) \end{aligned}

What is the circumference of a circle?

How to calculate the circumference of a circle

In order to calculate the circumference of a circle:

Find the radius or diameter of the circle.
Use the relevant formula to calculate the circumference of the circle.
Give your answer clearly with the correct units.

Explain how to calculate the circumference of a circle

Circumference of a circle worksheet

Get your free circumference of a circle worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Circumference of a circle worksheet

Get your free circumference of a circle worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Circumference of a circle examples

Example 1: calculating the circumference of the circle given the radius

A circle has a radius of $6cm$ .

Calculate its circumference.

Give your answer to $1$ decimal place

Find the radius or diameter of the circle.

The radius is given in the question

Radius $= 6cm$

2Use the relevant formula to calculate the circumference of the circle.

In this question you have the radius, therefore you should use the formula

\begin{aligned} &2\times\pi \times r\\\\ &2\times\pi \times 6\\\\ &12\pi \\\\ &37.699911... \end{aligned}

Remember the question asks you to round your answer to ‘1 decimal place’

37.7

3Give your answer clearly with the correct units.

Remember circumference is a measure of length, therefore the units should not be “squared”

37.7cm

Example 2: calculating the circumference of the circle given the diameter

A circle has a diameter of $10mm.$

Calculate its circumference.

Give your answer to $1$ decimal place

Find the radius or diameter of the circle.

In this question, the question gives you the diameter.

Diameter $= 10mm$

Use the relevant formula to calculate the circumference of the circle.

\begin{aligned} &\pi \times d\\\\ &\pi \times 10\\\\ &10\pi \\\\ &31.41592... \end{aligned}

Remember the question asks you to round your answer to ‘1 decimal place’

$31.4$

Give your answer clearly with the correct units.

31.4mm

Example 3: calculating the circumference of the circle, given the radius, answer in terms of 𝝅

A circle has a radius of $8m.$

Calculate its circumference.

Give your answer in terms of $\pi$

Find the radius or diameter of the circle.

The radius is given in the question

Radius $= 8m$

Use the relevant formula to calculate the circumference of the circle.

In this question you have the radius, therefore you should use the formula

$\begin{aligned} &2\times\pi \times r\\\\ &2\times\pi \times 8\\\\ &16\pi \end{aligned}$

Remember the question asks you to give your answer to ‘in terms of $\pi$ ‘.

Therefore you leave the answer in the form 16 $\pi$

Give your answer clearly with the correct units.

16\pi\; m

Example 4: calculating the circumference of the circle given the diameter

A circle has a diameter of $420km.$

Calculate its circumference.

Give your answer in terms of $\pi$

Find the radius or diameter of the circle.

In this question the question gives you the diameter.

Diameter $= 420km$

Use the relevant formula to calculate the circumference of the circle.

\begin{aligned} &\pi \times d\\\\ &\pi \times 420\\\\ &420\pi \end{aligned}

Remember the question asks you to give your answer to ‘in terms of $\pi$ .

Therefore you leave the answer in the form $420 \; \pi$

Give your answer clearly with the correct units.

420\pi \; km

Example 5: calculating the circumference of the circle given the area of a circle

A circle has a area of $21 cm^2.$

Calculate its circumference.

Give your answer to $2$ decimal places.

Find the radius or diameter of the circle.

The question gives you the area of the circle, but we need the radius/diameter in order to calculate the circumference.

We know that:

Area of a circle $= \pi r^2$

This means you can find the radius of the circle from the area, see below:

$\begin{aligned} \text { Area }&=\pi r^{2} \\\\ \text { Area }&=21 \\\\ 21&=\pi r^{2} \hspace{2.2cm} \text{Divide both sides by } \pi \\\\ \frac{21}{\pi}&=r^{2} \hspace{2.4cm} \text{Square Root both sides of the equation} \\\\ \sqrt{\frac{21}{\pi}}&=\sqrt{r^{2}} \\\\ 2.5854 \ldots&=r \end{aligned}$

Notice how we leave our answer to more decimals places than the question asks for.

This is so we do not cause a rounding error later on in the question.

Use the relevant formula to calculate the circumference of the circle.

In this question you now have the radius, therefore you should use the formula

$\begin{aligned} &2\times\pi \times r\\\\ &2\times\pi \times 2.5854\\\\ &16.24454.. \end{aligned}$

Remember the question asks you to round your answer to ‘2 decimal places’

$16.24$

Give your answer clearly with the correct units.

16.24cm

Example 6: calculating the perimeter of a semi-circle given the diameter

A semicircle has a diameter of $20m.$

Calculate its perimeter.

Give your answer in terms of $\pi$

Find the radius or diameter of the circle.

In this question the question gives you the diameter.

Diameter $= 20m$

Use the relevant formula to calculate the circumference of the circle.

\begin{aligned} &\pi \times d\\\\ &\pi \times 20\\\\ &20\pi \end{aligned}

Remember the question asks you to give your answer to ‘in terms of $\pi$ .

Therefore you leave the answer in the form $20 \pi$

$20\pi$ represents the circumference of a circle with a diameter of $20m$ . The perimeter of the semi circle is the length of the arc and the length of the diameter added together.

Length of arc

The length of the arc is half that of the whole circle because it is a semi circle.

Therefore the length of the arc is $20\pi$ divided by 2.

Length of arc $= 10\pi$

Length of diameter

Given to you in the question, $20m$

Circumference of a Circle example 6 step 2 1

Perimeter of circle

Length of arc + Length of diameter

$10\pi+20$

Remember to keep it in terms of $\pi$ as stated in the question

Give your answer clearly with the correct units.

(10\pi+20) m

Common misconceptions

Not including the correct units

When working with circumference you must always give the correct units, they should be not be squared.
E.g
$cm , m, km$ etc.

Not rounding correctly

These types of questions often involve rounding. You must only round at the end of the question and ensure you are rounding to what the question specifies.

Not giving answer in terms of $\pi$

Sometimes the question may ask you to give the answer ‘in terms of $\pi$ ’. This means you do not give the numerical answer that is produced when you multiply it by $\pi$

E.g.

6 x $\pi$ = 6 $\pi$ (this is an answer in terms of pi)

6 x $\pi = 18.8495592…$ (this answer is not in terms of pi)

Misuse of calculator

Ensure you know how to correctly use the $\pi$ button on your calculator

Circumference of a circle 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 circumference of a circle questions

18.8cm

6\pi cm^2

18.8m

6\pi cm

\begin{aligned} &\pi \times d \\ &= \pi \times 6 \\ &=6\pi cm \end{aligned}

18.8cm

6\pi cm^2

18.8m

6\pi cm

\begin{aligned} &\pi \times d \\ &= \pi \times 6 \\ &= 18.849… cm \end{aligned}

$18.8 cm$ (rounded to one decimal place)

31.4 \pi cm^2

20m^2

10\pi cm

24.5 cm^3

Circumference is a measure of length and this is the only answer with a measure of length $(cm)$ as its units.

\pi cm^2

\pi cm

6.3cm

6.28cm

\begin{aligned} &2 \times \pi \times r \\ &= 2 \times \pi \times 1 \\ &= 6.2831… cm \end{aligned}

$= 6.3 cm$ (rounded to one decimal place)

2\pi cm^2

2\pi cm

6.3cm

6.28cm

\begin{aligned} &2 \times \pi \times r \\ &= 2 \times \pi \times 1 \\ &= 2\pi cm \end{aligned}

This answer has been left in terms of $\pi$

4\pi cm

\pi cm

12.7cm

12.7 cm^2

\begin{aligned} & \pi \times d \\ &= \pi \times 4 \\ &= 4\pi cm \end{aligned}

628.3 cm^2

629 cm

628 cm

200\pi cm

\begin{aligned} & 2 \times \pi \times r \\ &= 2 \times\pi \times 100 \\ &= 628.3185.. cm \end{aligned}

$=628 cm$ (rounded to the nearest integer)

Remember integer means whole number

Circumference of a circle GCSE questions

1. The radius of a circle is $3.5 cm$ , work out the circumference of the circle.
Give your answer correct to $3$ significant figures.

(3 marks)

Show answer

$2 \times \pi \times3.5$ or $21.9911\dots$ seen

(1)

22.0

(1)

cm

(1)

2. The radius of a circle is $17.2 m$ , calculate the circumference of the circle.
Give your answer correct to $2$ decimal places.

(3 marks)

Show answer

$2 \times \pi \times17.2$ or $108.0707\dots$ seen

(1)

108.07

(1)

cm

(1)

3. The diameter of a circle is $12mm$ , work out the circumference of the circle.
Give your answer in terms of $\pi$ .

(2 marks)

Show answer

\pi \times 12

(1)

12\pi cm

(1)

4. The diameter of a circle is $34cm$ , work out the circumference of the circle.
Give your answer in terms of $\pi$ .

(2 marks)

Show answer

\pi \times 34

(1)

34\pi cm

(1)

5. The area of a circle is $72cm$ , work out the circumference of the circle.
Give your answer correct to the nearest integer.

(4 marks)

Show answer

$r = \sqrt{\frac{72}{\pi}}$ or $4.787\dots$ seen

(1)

2 \times \pi \times 4.787

(1)

30.0776\dots

(1)

30 cm

(1)

6. A tile is in the shape of a semicircle has a perimeter of $15cm$ .
Work out the length of the diameter. Give your answer correct to one decimal place.

(5 marks)

Show answer

Seen or implied that the perimeter of a semicircle is the diameter added to the length of the arc. Could be seen in a diagram.

E.g.

Diameter + half of the circumference $= 15$

(1)

Seen or implied that the length of the arc half of the total circumference

E.g.

$\frac{\pi d}{2}$ seen

(1)

Correct method to find the diameter of the circle e.g.

d = \frac{15}{1+\frac{\pi}{2}}

(1)

5.834\dots

(1)

5.8 cm

(1)

7. The circumference of a circle is $64\pi cm$ .
Calculate the radius of the circle.

(2 marks)

Show answer

64\pi = 2\times\pi \times r

(1)

32 = r

(1)

Learning checklist

You have now learned how to:

Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference

Know the formulae for circumference of a circle

Give answers in terms of

\pi

Calculate circumference of circles and the perimeter semi-circles

The next lessons are

Still stuck?

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