GCSE Maths Geometry and Measure

Area

Area of a Circle

# Area Of A Circle

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

There are also area 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 area of a circle?

The area of a circle is given by the area of a circle formula which is made by using a specific relationship between the radius of a circle and its area.

Area of a circle formula:

\pi \times r\times r

we usually simplify this to

E.g.

What is the area of a circle with radius 3cm ?

\begin{aligned} \text { Area } &=\pi r^{2} \\\\ &=\pi \times 3^{2} \\\\ &=9 \pi \mathrm{cm}^{2} \\\\ &=28.3 \mathrm{~cm}^{2}(1 . \mathrm{d} . \mathrm{p}) \end{aligned}

### What is pi?

\pi (pronounced pi) represents the ratio of the circumference of a circle to its diameter. For all circles if you divide the length of the circumference by the length of the diameter you get the value \pi .

E.g

If the diameter of a circle was 1m , its circumference would be \pi \; m.

Note: \pi is an irrational number which means it cannot be written as a fraction, and in a non recurring decimal has an approximate value of 3.14159…

In GCSE you should use the \pi button on your calculator when working with \pi . To get this you need to press [SHIFT][ \times 10^x ].

If your calculator does not allow you to do this you can use the value 3.141*

Sometimes the question may ask you to give the answer ‘in terms of \pi ’.

This means you do not give the numerical answer when you multiply it by \pi .

E.g.

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

8 \times \pi = 8\pi (this is an answer in terms of pi)

10 \times \pi =  31.41... (this is an answer not in terms of pi)

## How to calculate the area of a circle

In order to calculate the area of a circle:

1. Find the radius of the circle.
2. Use the formula \text{Area of a circle} = \pi r^2 to calculate the area of the circle.
3. Give your answer clearly with the correct units.

## Area of a circle examples

### Example 1: calculating the area of the circle given the radius

A circle has a radius of 6cm .

Calculate its area.

1. Find the radius of the circle.

The radius is given in the question

2Use the formula \pi r^2 to calculate the area of the circle.

\begin{aligned} &\pi r^2\\\\ &= \pi \times r\times r\\\\ &= \pi \times6\times 6\\\\ &= 36\pi \\\\ &= 113.0973355... \end{aligned}

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

113.10

3Give your answer clearly with the correct units.

113.10cm^2

### Example 2: calculating the area of the circle given the diameter

A circle has a diameter of 10mm.

Calculate its area.

In this question the question gives you the diameter. You need the radius to find the area of the circle.

Remember the diameter of the circle is twice the radius.

Diameter = 10mm

Radius =10mm \div 2

\begin{aligned} &\pi r^2\\\\ &= \pi \times r\times r\\\\ &= \pi \times5\times 5\\\\ &= 25\pi \\\\ &=78.53981... \end{aligned}

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

78.5

78.5mm^2

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

A circle has a radius of 8m.

Calculate its area.

Give your answer in terms of \pi

The radius is given in the question

\begin{aligned} &\pi r^2\\\\ &= \pi \times r\times r\\\\ &= \pi \times8\times 8\\\\ &= 64\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 64 \pi

64\pi m^2

### Example 4: calculating the area of the circle given the diameter

A circle has a diameter of 420km.

Calculate its area.

Give your answer in terms of \pi

In this question the question gives you the diameter. You need the radius to find the area of the circle.

Diameter = 420km

Radius = 420km \div 2

\begin{aligned} &\pi r^2\\\\ &= \pi \times r\times r\\\\ &= \pi \times210\times 210\\\\ &= 44100\pi \end{aligned}

You leave the answer in the form 44100 \; \pi

44100\pi km^2

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

A circle has a circumference of 21cm.

Calculate its area.

The question gives you the circumference of the circle. But you need the radius

You know that the circumference of a circle is equal to 2\pi r

Step by step guide: Circumference of a circle

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

Circumference = 2\pi r

Circumference = 21

21 = 2\pi r

Divide both sides by 2\pi

\frac{21}{2\pi}= r

Notice how we leave our answer in terms of \pi at this stage. This is so you do not cause a rounding error later on in the question.

\begin{aligned} &\pi r^2\\\\ &= \pi \times r\times r\\\\ &= \pi \times\frac{21}{2\pi}\times \frac{21}{2\pi}\\\\ &=35.09366.. \end{aligned}

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

35.09

35.09cm^2

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

A semicircle has a diameter of 20m.

Calculate its area.

Give your answer in terms of \pi

In this question the question gives you the diameter. You need the radius to find the area of the circle.

Remember the diameter of the circle is twice the radius.

Diameter = 20m

Radius = 20m \div 2

\begin{aligned} &\pi r^2\\\\ &= \pi \times r\times r\\\\ &= \pi \times 10\times 10\\\\ &= 100\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 100 \pi

100\pi

Represents the area of a whole circle with a diameter of 20m. You only want the area of a semi circle. A semi circle has half the area of a full circle so you need to divide your answer by two. Remember to keep it in terms of \pi .

\begin{aligned} &100\pi \div2 \\\\ &50\pi \end{aligned}

50 \pi m^2

### Common misconceptions

• Not using the radius

You must have the radius to find the area of a circle from the formula.

A question may not give you the radius directly, for example it may give you the diameter. You must use the information given to find the radius first

• Not including the correct units

When working with area you must always give the correct units squared

E.g cm^2 , m^2, km^2 etc.

• Not rounding correctly

These questions often involve rounding. You must only round at the end of the question and ensure you are rounding to what the question specifies e.g to 2 decimal places

•  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

### Practice area of a circle questions

1. A circle has a diameter of 6cm . What is the radius of the circle?

6cm

3cm

12cm

6\pi cm

The diameter of the circle is twice the size of the radius. Therefore to find the radius you can divide the diameter by 2.

6cm \div 2 = 3cm

2. Which of these answers is in terms of \pi ?

31.4cm

20m

10\pi cm

24.5

10 \pi means 10 lots of \pi

3. A circle has a radius of 1cm. What is its area to 1 decimal place?

\pi cm^2

\pi cm

3.1cm

3.1 cm^2

Area of a circle = \pi r^2

\pi \times 1 \times 1 is equal to 3.1415…

This answer is correctly rounded to 1 decimal place and has the correct units.

4. A circle has a radius of 1cm. What is its area in terms of \pi ?

\pi cm^2

\pi cm

3.1cm

3.1 cm^2

Area of a circle = \pi r^2

\pi \times 1 \times 1 is equal to \pi

This answer is correctly given in terms of \pi and has the correct units.

5. A circle has a diameter of 2cm. What is its area in terms of \pi ?

4\pi cm^2

\pi cm^2

3.1cm

3.1 cm^2

You must first divide the diameter by 2 to find the radius, 2cm divided by 2 is equal to 1cm.

Therefore the radius is 1cm

Area of a circle = \pi r^2

\pi \times 1 \times 1 is equal to \pi

This answer is correctly given in terms of \pi and has the correct units.

6. A circle has a diameter of 100cm. What is its area to the nearest whole number?

31416 cm^2

31415 cm^2

31415.9 cm^2

10000\pi cm^2

Area of a circle = \pi r^2

\pi \times 100 \times 100 is equal to 31415.92654…

Which is 31416 cm^2 rounded to the nearest whole number.

### Area of a circle GCSE questions

1. The radius of a circle is 3.5 cm

Work out the area of the circle

Give your answer correct to 3 significant figures

(3 marks)

\pi \times3.5\times 3.5   or  38.4845… seen

(1)

38.5

(1)

cm^2

(1)

2. The radius of a circle is 17.2 m

Work out the area of the circle

Give your answer correct to 2 decimal places

(3 marks)

\pi \times17.2\times 17.2   or  929.408… seen

(1)

929.41

(1)

m^2

(1)

3. The diameter of a circle is 20mm

Work out the area of the circle

Give your answer in terms of \pi

(3 marks)

\pi \times10\times 10

(1)

100\pi

(1)

mm^2

(1)

4. The diameter of a circle is 15cm

Work out the area of the circle

Give your answer in terms of \pi

(3 marks)

\pi \times7.5\times 17.5

(1)

56.25\pi

(1)

cm^2

(1)

5. The circumference of a circle is 72cm

Work out the area of the circle

Give your answer correct to the nearest integer

(4 marks)

\frac{72}{2\pi}= r   or    \frac{36}{\pi}= r seen

(1)

\pi \times\frac{72}{2\pi}\times \frac{72}{2\pi}

(1)

412.52961… seen

(1)

413

(1)

6. A tile is in the shape of a semicircle

The perimeter of a semi circle is 12.85cm

The length of the arc is 7.85cm

Work out the total area of the tile

Give your answer correct to the nearest integer

(4 marks)

Length of diameter = 5cm   could be implied in working out or seen on diagram

(1)

\pi \times 2.5\times 2.5

6.25\pi or 19.634… seen

(1)

“19.634…” ÷ 2 or 9.817… seen

(1)

10

(1)

7. The area of a circle is 64\pi cm^2

Calculate the radius of the circle

(3 marks)

64\pi = \pi r^2

(1)

64 = r^2

(1)

r=8

(1)

Only positive value should be given

## 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 area of a circle
• Give answers in terms of \pi
• Calculate area of 2D shapes including circles and semi-circles

## The next lessons are

• Circumference of a circle
• Arc lengths
• Area of a sector
• Perimeter of a sector
• Circle graph
• Equation of a circle
• Circle theorems
• Surface area and volume of spheres

## Still stuck?

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