One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

Learn more
GCSE Maths Geometry and Measure 3D Shapes Sphere

Surface Area Of A Sphere

Surface Area Of A Sphere

Here we will learn about the surface area of a sphere, including how to calculate the surface area of a sphere given its radius and the surface area of a hemisphere.  We will look at some practice problems.

There are also surface area of a sphere 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 surface area of a sphere?

The surface area of a sphere is the area which covers the outer surface of a sphere.

Surface area of a sphere Image 1

The formula for the surface area of a sphere with radius r r is:

Surface area=4πr2 \text{Surface area}=4\pi{r}^{2}

Notice that the square of the radius (r2) (r^2) occurs within the surface area formula.

The surface area of a shape uses two-dimensions so the units for surface area are units squared.

What is the surface area of a sphere?

What is the surface area of a sphere?

How to calculate the surface area of a sphere

In order to calculate the surface area of a sphere:

  1. Write down the formula.
  2. Substitute the given values into the formula.
  3. Complete the calculation.
  4. Write the final answer, including the units.

How to calculate the surface area of a sphere

How to calculate the surface area of a sphere

Surface area of a sphere worksheet

Surface area of a sphere worksheet

Surface area of a sphere worksheet

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

DOWNLOAD FREE
x
Surface area of a sphere worksheet

Surface area of a sphere worksheet

Surface area of a sphere worksheet

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

DOWNLOAD FREE

Related lessons on sphere

Surface area of a sphere is part of our series of lessons to support revision on sphere. You may find it helpful to start with the main sphere 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:

Surface area of a sphere examples

Example 1: surface area of a sphere given the radius

Find the surface area of the sphere below. Write your answer to 1 1 decimal place.

Surface area of a sphere Example 1

  1. Write down the formula.

To answer the question we need to use the formula for the surface area of a sphere:

Surface area=4πr2 \text{Surface area}=4 \pi r^2

2Substitute the given values into the formula.

The radius (the distance from the centre of the sphere to the surface) is 4.8  cm. 4.8 \; cm. We need to substitute the value of the radius into the formula for the surface area of a sphere:

SA =4×π×4.82 \text{SA }=4 \times \pi \times 4.8^2

3Complete the calculation.

Use a calculator to work out the value for the surface area:

SA =4×π×4.82=230425π=289.529179 \begin{aligned} \text{SA }&=4 \times \pi \times 4.8^2\\\\ &=\frac{2304}{25}\pi\\\\ &=289.529179…… \end{aligned}

4Write the final answer, including the units.

Check what form the final answer needs to be. Here we are asked to give the answer to 1 1 decimal place.

289.529179=289.5 cm2 (1dp) 289.529179… = 289.5 \text{ cm}^{2} \ \text{(1dp)}

The surface area of the sphere is 289.5cm2  (1dp). 289.5cm^2 \; (1dp).

Example 2: surface area of a sphere

Find the surface area of a sphere with the diameter of 42  mm. 42 \; mm.

Write your answer to the nearest square millimetre.

Surface area of a sphere Example 2 new

Write down the formula.

Show step

Substitute the given values into the formula.

Show step

Complete the calculation.

Show step

Write the final answer, including the units.

Show step

Example 3: surface area of a sphere – in terms of π

Find the surface area of a sphere with a radius of 6  cm. 6 \; cm.

Leave your answer in terms of π.

Surface area of a sphere Example 3

Write down the formula.

Show step

Substitute the given values into the formula.

Show step

Complete the calculation.

Show step

Write the final answer, including the units.

Show step

Surface area of a hemisphere

A hemisphere is half of a sphere although the surface area of a hemisphere is not half of the surface area of a sphere. This is because we have an extra circular face that we have to add onto the surface area of the curved face.

As the surface area of a sphere is equal to 4πr2 4\pi{r}^{2} , half of this formula would obtain the curved surface area of a hemisphere (the dome). The surface area of the dome is therefore 2πr2 2\pi{r}^{2} . A circle has an area of πr2 \pi{r}^{2} and because we are talking about the same 3 3 D shape, the radius r r is the same. If we add the area of the two surfaces together, we get the surface area of a hemisphere to be:

 SA=2πr2+πr2=3πr2 \begin{aligned} \text{ SA}&=2\pi{r}^{2}+\pi{r}^{2}\\\\ &=3\pi{r}^{2} \end{aligned}

Surface area of a hemisphere =3πr2 \text{Surface area of a hemisphere }=3\pi{r}^{2}

for a hemisphere with radius r r .

Surface area of a hemisphere examples

Example 4: surface area of a hemisphere

Calculate the total surface area of a hemisphere with radius 7  cm 7 \; cm . Write your answer to 1 1 decimal place.

Surface area of a sphere Example 4

Write down the formula.

Show step

Substitute the given values into the formula.

Show step

Complete the calculation.

Show step

Write the final answer, including the units.

Show step

Example 5: surface area of a hemisphere given the diameter

Calculate the total surface area of a hemisphere with a diameter of 26.4  cm 26.4 \; cm . Write your answer in terms of π \pi .

Surface area of a sphere Example 5 new

Write down the formula.

Show step

Substitute the given values into the formula.

Show step

Complete the calculation.

Show step

Write the final answer, including the units.

Show step

Example 6: radius of a sphere

The surface area of a sphere is 4000  cm2 4000 \; cm^2 . Calculate the radius of the sphere. Write your answer to 1 1 decimal place.

Write down the formula.

Show step

Substitute the given values into the formula.

Show step

Complete the calculation.

Show step

Write the final answer, including the units.

Show step

Common misconceptions

  • Using the correct formula

There are several formulas that can be used, so we need to match the correct formula to the correct context.

  • Rounding

It is important to not round the answer until the end of the calculation. This will mean your final answer is accurate. 

  • Make sure you have the correct units

For area we use square units such as cm2 cm^2 .  

For volume we use cube units such as cm3 cm^3 .

  • Using the radius or the diameter

It is a common error to mix up radius and diameter. Remember the radius is half of the diameter.

  • Confusing the volume formula with the surface area formula

It is a common error to mix up the formula for the volume of the sphere with the formula for the surface area of the sphere.

Remember that the surface area formula includes a squared term, whereas the volume formula includes a cubed term.

Practice surface area of a sphere questions

1. Find the surface area of a sphere of radius 10  cm 10 \; cm
Give your answer to 1 1 decimal place.

 

Surface area of a sphere Praction Question 1

314.2 cm2 314.2 \ cm^2
GCSE Quiz False

942.5 cm2 942.5 \ cm^2
GCSE Quiz False

1256.6 cm2 1256.6 \ cm^2
GCSE Quiz True

4188.8 cm2 4188.8 \ cm^2
GCSE Quiz False

We are finding the surface area of a sphere, so we substitute the value of r r into the formula.

 

Surface area=4πr2=4×π×102=400π=1256.637061=1256.6 cm2 (1dp) \begin{aligned} \text{Surface area}&=4\pi r^2\\\\ &=4 \times \pi \times 10^2\\\\ &=400\pi\\\\ &=1256.637061…\\\\ &=1256.6 \text{ cm}^2 \ (1dp) \end{aligned}

2. Find the surface area of a sphere of diameter 6.4  cm 6.4 \; cm
Write your answer to 1 1 dp.

 

Surface area of a sphere Practice Question 2

32.2 cm2 32.2 \ cm^2
GCSE Quiz False

128.7 cm2 128.7 \ cm^2
GCSE Quiz True

137.3 cm2 137.3 \ cm^2
GCSE Quiz False

514.7 cm2 514.7 \ cm^2
GCSE Quiz False

We are finding the surface area of a sphere, so we substitute the value of r r into the formula. We have been given the diameter, so we need to halve 6.4 6.4 to get the radius, then substitute this value into the formula for the surface area of a sphere.

 

r=d÷2=6.4÷2=3.2 cm. r=d\div{2}=6.4\div{2}=3.2\text{ cm}.

 

Surface area=4πr2=4×π×3.22=102425π=128.679351=128.7 cm2 (1dp) \begin{aligned} \text{Surface area}&=4\pi r^2\\\\ &=4 \times \pi \times 3.2^2\\\\ &=\frac{1024}{25}\pi\\\\ &=128.679351…\\\\ &=128.7 \text{ cm}^2 \ (1dp) \end{aligned}

3. Calculate the surface area of a sphere of radius 9  cm 9 \; cm . Leave your answer in terms of π \pi .

 

Surface area of a sphere Practice Question 3

324π cm2 324\pi \text{ cm}^2
GCSE Quiz True

36π cm2 36\pi \text{ cm}^2
GCSE Quiz False

81π cm2 81\pi \text{ cm}^2
GCSE Quiz False

972π cm2 972\pi \text{ cm}^2
GCSE Quiz False

We are finding the surface area of a sphere, so we substitute the value of r r into the formula.

 

Surface area=4πr2=4×π×92=324π=324π cm2 \begin{aligned} \text{Surface area}&=4\pi r^2\\\\ &=4 \times \pi \times 9^2\\\\ &=324\pi\\\\ &=324\pi \text{ cm}^2 \end{aligned}

4. Find the total surface area of a hemisphere with radius 8  cm 8 \; cm
Write your answer to 1 1 dp.

 

Surface area of a sphere Practice Question 4

201.0 cm2 201.0 \ cm^2
GCSE Quiz False

201.1 cm2 201.1 \ cm^2
GCSE Quiz False

603.1 cm2 603.1 \ cm^2
GCSE Quiz False

603.2 cm2 603.2 \ cm^2
GCSE Quiz True

The surface area of a hemisphere is equal to 3πr2 3\pi{r}^{2} . Substituting r=8  cm r=8 \; cm into the formula, we have

 

SAH =3×π×82=192π=603.1857895=603.2 cm2 (1dp) \begin{aligned} \text{SAH }&=3\times\pi\times{8}^2\\\\ &=192\pi\\\\ &=603.1857895…\\\\ &=603.2\text{ cm}^2 \ (1dp) \end{aligned}

5. Find the total surface area of a hemisphere with radius 12  cm 12 \; cm . Leave your answer in terms of π \pi .

 

Surface area of a sphere Practice Question 5

432πcm2 432\pi \text{cm}^2
GCSE Quiz True

144π cm2 144\pi \text{ cm}^2
GCSE Quiz False

720π cm2 720\pi \text{ cm}^2
GCSE Quiz False

576π cm2 576\pi \text{ cm}^2
GCSE Quiz False

The surface area of a hemisphere is equal to 3πr2 3\pi{r}^{2} . Substituting r=12  cm r=12 \; cm into the formula, we have

 

SAH =3×π×82=432π \begin{aligned} \text{SAH }&=3\times\pi\times{8}^2\\\\ &=432\pi \end{aligned}

6. The surface area of a sphere is 9000  cm2. 9000 \; cm^2. Calculate the radius of the sphere. Write your answer correct to 3 3 significant figures.

26.7 cm 26.7 \ cm
GCSE Quiz False

26.8 cm 26.8 \ cm
GCSE Quiz True

53.5 cm 53.5 \ cm
GCSE Quiz False

53.6 cm 53.6 \ cm
GCSE Quiz False

Using the formula we substitute the value of the surface area in and rearrange to find the radius.

 

SA =4πr29000=4πr22250=πr22250π=r2r=2250πr=26.76186174r=26.8  cm  (3sf) \begin{aligned} \text{SA }&=4 \pi r^2 \\\\ 9000 &= 4\pi r^2 \\\\ 2250 &= \pi r^2\\\\ \frac{2250}{\pi}&=r^2\\\\ r&=\sqrt{\frac{2250}{\pi}} \\\\ r&=26.76186174…\\\\ r&=26.8 \; cm \; (3sf) \end{aligned}

Surface area of a sphere GCSE questions

Surface area of a sphere=4πr2 \text{Surface area of a sphere}=4 \pi r^2

1. Here is a sphere with a radius 11  cm. 11 \; cm.

 

Surface area of a sphere GCSE Question 1

 

Calculate the surface area of the sphere. Write your answer to 2 2 decimal places. State the units in your answer.

 

(3 marks)

Show answer
=4×π×112 =4\times \pi \times 11^2

(1)

 

=484π=1520.530844 =484\pi=1520.530844…

(1)

 

1520.53  cm2 1520.53 \; cm^2

(1)

2. Here is a hemisphere with a radius 5  mm. 5 \; mm.

 

surface area of a sphere gcse question 2 new

 

Calculate the total surface area of the hemisphere. Give your answer to 3 3 significant figures. State the units in your answer.

 

(3 marks)

Show answer
SAH =3πr2=3×π×52 \text{SAH }=3\pi{r}^2=3\times\pi\times{5}^2

(1)

 

SAH =75π=235.619449 \text{SAH }=75\pi=235.619449…

(1)

 

236  mm2 236 \; mm^2

(1)

3. A spherical ball has a radius of 30  cm. 30 \; cm.

 

surface area of a sphere gcse question 3 150x150

 

A tin of paint covers 2000  cm2. 2000 \; cm^2.

A tin of paint costs £13. £13.

Calculate how much it would cost to paint the outside surface of the spherical ball.

 

(3 marks)

Show answer
4×π×302=3600π=11309.73355 4 \times \pi \times 30^2=3600\pi=11309.73355…

(1)

 

11309.73355÷2000=5.654866776=6 tins 11309.73355… \div {2000} = 5.654866776… = 6 \text{ tins}

(1)

 

6×13=£78 6 \times 13=\pounds{78}

(1)

Learning checklist

You have now learned how to:

  • calculate surface areas of spheres and composite solids

The next lessons are

Did you know?

Archimedes was a famous ancient Greek mathematician who lived about 2,200 2,200 years ago in Sicily.  He wrote about the surface area of a sphere and said it was four times the area of the greatest circle.  This is the same formula that we use today.  

Archimedes also worked on cylinders and discovered the formula for the lateral surface area of a cylinder.  The lateral surface area excludes its circular base and top.  We would usually refer to this as its curved surface area.

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

GCSE Benefits

Find out more about our GCSE maths tuition programme.