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Here we will learn about calculating the area and circumference of a circle.

There are also area and 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.

The** area of a circle** is the amount of space within a circle and the **circumference of a circle** is the distance around the edge of the circle.

To calculate the area and circumference of a circle we use the following circle formulae,

\text{Area }= \pi r^{2} \text{Circumference }= \pi d = 2\pi rwhere r is the radius of the circle (the distance from the centre of the circle to the edge) and d is the diameter of the circle (the distance across the circle passing through the centre of the circle).

The symbol \pi is the Greek letter “pi” where the value of pi is 3.14159265358979…

As \pi is an irrational number, the approximate value of \pi is 3.142 \ (3dp).

The circumference of the circle is the perimeter of the circle and is proportional to its radius; as the radius increases, the circumference increases.

The circumference is a length and so it is measured using units such as mm, \ cm, \ m.

Area is measured in square units, such as mm^2, \ cm^2 or m^2.

In order to calculate the area or circumference of a circle:

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

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

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

DOWNLOAD FREE**Area and circumference of a circle** is part of our series of lessons to support revision on **parts of a circle** and **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:

Find the area of this circle. Give your answer to 1 decimal place.

**Find the radius or diameter of the circle.**

To find the area, we need to know the radius. The radius of this circle is 9 \ cm.

2**Use the relevant formula.**

The circle formula for area is A= \pi r^{2}.

\begin{aligned} A&= \pi r^{2}\\\\ A&= \pi \times 9^{2}\\\\ &=254.4690049 \end{aligned}3**Give your answer clearly with the correct units.**

We need to give our answer to 1 decimal place. Since the radius is measured in cm, the area will be measured in cm^2.

\text{Area }=254.5cm^{2}\text{ (1dp)}Find the area of this circle. Give your answer in terms of \pi .

**Find the radius or diameter of the circle.**

To find the area, we need to know the radius. The diameter of this circle is 6 \ mm and therefore the radius is 3 \ mm.

**Use the relevant formula.**

The formula for area is A= \pi r^{2}.

\begin{aligned}
A&= \pi r^{2}\\\\
A&= \pi \times 3^{2}\\\\
&=9 \pi
\end{aligned}

**Give your answer clearly with the correct units.**

Here we are asked to give our answer in terms of pi (\pi). The diameter is measured in mm so the area will be measured in mm^2.

\text{Area }= 9 \pi \ {mm}^{2}

Find the circumference of this circle. Give your answer to 3 significant figures.

**Find the radius or diameter of the circle.**

To find the circumference we can use the radius or the diameter. The radius of this circle is 21 \ m.

**Use the relevant formula.**

To find the circumference using the radius we can use the formula C=2 \pi r.

\begin{aligned}
C&=2 \pi r\\\\
C&=2 \times \pi \times 21\\\\
& =131.9468915
\end{aligned}

**Give your answer clearly with the correct units.**

Here we are asked to give our answer to 3 significant figures. The radius is measured in m so the circumference will also be measured in m.

\text{Circumference }=132m\text{ (3sf)}

Find the circumference of this circle. Give your answer in terms of \pi .

**Find the radius or diameter of the circle.**

To find the circumference we can use the radius or the diameter. The diameter of this circle is 17 \ cm.

**Use the relevant formula.**

To find the circumference using the diameter we can use the formula C= \pi d.

\begin{aligned}
C&= \pi d\\\\
C&=\pi \times 17\\\\
&=17 \pi
\end{aligned}

**Give your answer clearly with the correct units.**

Here we are asked to give the answer in terms of \pi. The diameter is measured in cm so the circumference will also be measured in cm.

\text{Circumference }=17 \pi ~ \mathrm{cm}

Find the area of this semicircle. Give your answer to 2 decimal places.

**Find the radius or diameter of the circle.**

From the diagram of the semicircle, we can see that the diameter of the circle is 10 \ cm. For area we need the radius which is 10 \div 2=5 \ cm.

**Use the relevant formula.**

To find the area of a semicircle, we find the area of the whole circle and half it.

\begin{aligned}
A&=\pi{r}^{2}\div{2}\\\\
&=\pi\times{5}^{2}\div{2}\\\\
&=25\pi\div{2}\\\\
&=12.5\pi\\\\
&=39.26990817
\end{aligned}

**Give your answer clearly with the correct units.**

We have been asked to give the answer to 2 decimal places. The diameter is measured in cm therefore the area will be measured in cm^2.

\text{Area }=39.27 \ cm^{2}\text{ (2dp)}

The area of a circle is 30 \ cm^2. Work out the circumference of the circle. Give your answer to 3 significant figures.

**Find the radius or diameter of the circle.**

In this case we are given the area of the circle and we need to work backwards to find the radius. Substituting the values we know into the area formula we get,

\begin{aligned} \text{Area }&= \pi r^{2}\\\\ 30&= \pi r^{2} \end{aligned}

We can then solve this to find r.

30=\pi{r}^{2}

Dividing both sides by \pi, we get

\frac{30}{\pi}=r^{2}.

As we have r^2, we need to square root both sides to get r on its own.

\sqrt{\frac{30}{\pi}}=r

Evaluating this, we have r=3.090193616.

It is important not to round this value yet as this will cause us to lose accuracy in our final answer.

**Use the relevant formula.**

Now that we have the radius we can use the relevant formula to find the circumference.

\begin{aligned}
C&=2 \pi r\\\\
&=2 \times \pi \times 3.090193616\\\\
&=19.41625913
\end{aligned}

**Give your answer clearly with the correct units.**

We have been asked to give the answer to 3 significant figures. The area is measured in cm^2 so the circumference will be measured in cm.

\text{Circumference }=19.4 \ cm \ (3sf)

**Incorrectly using the radius or diameter of a circle**

It is important to know the difference between the diameter and the radius of a circle and to use the correct one in the calculation.

**Using an incorrect formula**

It can be easy to get confused between the different formulae – make sure you know which formula is for which calculation.

**Incorrect use of BIDMAS**

When calculating the area of a circle, we need to calculate the square of the radius first as the radius is raised to the power of 2, and then multiply this value by \pi . This is because indices come before multiplication in BIDMAS.

For example, for a circle with radius 4 \ cm,

A=\pi\times{4}^{2}=\pi\times{16}=50.26548246… (Correct)

A=(\pi\times{4})^{2}=\pi^{2}\times{16}=157.9136704… (Incorrect)

**Not including the correct units**

Remember that circumference is a measure of length, such as mm, \ cm or m and area is measured in square units, such as mm^2, \ cm^2 or m^2.

**Not rounding correctly**

Read the question again at the end to ensure you have given the answer to the correct degree of accuracy and always ensure you apply the rules of rounding when rounding an answer.

1. Find the area of the circle. Give your answer to 1 decimal place.

69.1 \ cm^2

1520.5 \ cm^2

34.6 \ cm^2

380.1 \ cm^2

The radius of the circle is 11 \ cm.

\begin{aligned} \text{Area }&=\pi r^{2}\\\\ &=\pi \times 11^{2}\\\\ &=380.1327111 \\\\ &=380.1 \mathrm{cm}^{2}~\text{(1dp)} \end{aligned}

2. Find the area of the circle. Give your answer in terms of \pi.

49 \pi \ \mathrm{mm}^{2}

14 \pi \ \mathrm{mm}^{2}

28 \pi \ \mathrm{mm}^{2}

196 \pi \ \mathrm{mm}^{2}

The diameter is 14 \ mm so the radius is 7 \ mm.

\begin{aligned} \text{Area }&= \pi r^{2}\\\\ &= \pi \times 7^{2}\\\\ &=49 \pi ~ \mathrm{mm}^{2} \end{aligned}

3. Find the circumference of the circle. Give your answer to 3 significant figures.

2.51 \ m

2.01 \ m

5.03 \ m

8.04 \ m

The radius of the circle is 0.8m.

\begin{aligned} \text{Circumference }&=2 \pi r\\\\ &= 2 \times \pi \times 0.8\\\\ &=5.026548246\\\\ &=5.03 \ \mathrm{m} ~ \text{(3sf)} \end{aligned}

4. Find the circumference of the circle. Give your answer in terms of \pi.

26 \pi ~\mathrm{cm}

13 \pi ~\mathrm{cm}

169 \pi ~\mathrm{cm}

676 \pi ~\mathrm{cm}

The diameter of the circle is 26m.

\begin{aligned} \text{Circumference}&=\pi d\\\\ &=\pi \times 26\\\\ &=26 \pi~\mathrm{cm} \end{aligned}

5. Find the area of this quarter circle. Give your answer to 3 significant figures.

1020 \ cm^2

254 \ cm^2

28.3 \ cm^2

63.3 \ cm^2

Here we want to find the area of \frac{1}{4} of a circle. We can see from the diagram that the radius of the circle is 18 \ cm. We can find the area of the whole circle and then divide by 4.

\begin{aligned} \text{Area of whole circle }&=\pi r^{2}\\\\ &=\pi \times 18^{2}\\\\ &=324\pi \\ \end{aligned}

\begin{aligned} \text{Area of quarter circle }= 324\pi \div 4 &= 81\pi\\\\ &=254.4690049\\\\ &=254\mathrm{cm}^{2}~\text{(3sf)} \end{aligned}

6. The circumference of a circle is 50 \ cm. Work out the area of the circle. Give your answer to 2 decimal places.

1963.50 \ cm^2

15.92 \ cm^2

795.77 \ cm^2

198.94 \ cm^2

\begin{aligned}
\text{Circumference }&=\pi d\\\\
50&=\pi d\\\\
\frac{50}{\pi}&=d\\\\
d&=15.91549431
\end{aligned}

\begin{aligned} r&=15.91549431 \div 2\\\\ &=7.957747155 \end{aligned}

\begin{aligned} \text{Area }&=\pi r^{2}\\\\ &=\pi \times 7.957747155^{2}\\\\ &=198.9436789\\\\ &=198.94 \ \mathrm{cm}^{2} \ (2dp) \end{aligned}

1. Josh is calculating the circumference of this circle.

Here is Josh’s working.

\begin{aligned} 2 \times \pi \times 22 &=138.2300768\\\\ &=138.2 \ \mathrm{cm}~\text{(1pd)} \end{aligned}

What mistake has Josh made?

**(1 mark)**

Show answer

He has used the diameter (22 \ cm) instead of the radius (11 \ cm) in C=2 \pi r.

**(1)**

2. (a) Vicky has a garden with a circular pond, as shown below.

Calculate the area of the pond. Give your answer to 3 significant figures.

(b) Vicky wants to buy grass seed to cover the lawn.

Grass seed comes in boxes which cover 20 \ m^2 per box.

Vicky has 5 boxes of seed. Does she have enough to cover her lawn?

**(5 marks)**

Show answer

(a)

Radius =2.5 \ m

**(1)**

**(1)**

(b)

Area of the garden is 8 \times 14=112 \ \mathrm{m}^{2} .

**(1)**

Area of the lawn is 112-19.6=92.4 \ \mathrm{m}^{2} .

**(1)**

92.4 \div 20=4.62 boxes required. Yes she does.

**(1)**

3. Here are two pizzas sold in a restaurant. Which pizza is better value for money?

**(5 marks)**

Show answer

Area of the large pizza is \pi \times 18^{2}=1017.876 \ \mathrm{cm}^{2} .

**(1)**

1017.876 \div 12 = 84.82 \ \mathrm{cm}^{2} per £1 .

**(1)**

Area of the small pizza is \pi\times 14^{2}=615.752 \ \mathrm{cm}^{2} .

**(1)**

615.75 \div 8=76.97 \ \mathrm{cm}^{2} per £1 .

**(1)**

Large pizza is better value.

**(1)**

You have now learned how to:

- Calculate the area of a circle
- Calculate the circumference of a circle
- Solve problems involving area and circumference of a circle

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