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

3D trigonometry

Calculating area

Substituting into formulae

Rounding

This topic is relevant for:

GCSE Maths Geometry and Measure

Surface Area

How to Calculate Surface Area

Here we will learn how to calculate the surface area of a variety of three-dimensional shapes, including cuboids, prisms, cylinders, cones and spheres.

There are also surface area 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 surface area?

The surface area of a three dimensional shape is the total area of all of the faces.

To find the surface area of a shape, we find the area of each face and add them together.

E.g.

Face	Area
Front	½ × 4 × 3 = 6
Back	6
Bottom	4 × 6 = 24
Top	5 × 6 = 30
Left side	3 × 6 = 18

Total surface area

\begin{aligned} &= 6+6+24+30+18 \\ &=84cm^2 \end{aligned}

The measurements here are in centimetres so the surface area is measured in square centimetres $(cm^2)$ .

What is surface area?

How to calculate surface area

In order to calculate surface area:

Calculate the area of each face.
Add the areas together.
Write the answer, including the units.

Explain how to calculate surface area

Surface area worksheet

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

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Surface area worksheet

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

DOWNLOAD FREE

Surface area examples

Example 1: surface area of a cuboid

Work out the surface area of the cuboid. We can think of this as finding the surface area of a rectangular prism or the surface area of a box.

Calculate the area of each face.

Face	Area (cm²)
Front	10 × 4 = 40
Back	40
Bottom	10 × 6 = 60
Top	60
Left side	6 × 4 = 24
Right side	24

2Add the areas together.

The sum of the areas is: $40+40+60+60+24+24=248$

3Write the answer, including the units.

The measurements are in $cm$ so the surface area will be measured in $cm^2$ .

Total surface area $= 248cm^2$

Example 2: surface area of a cube

Work out the surface area of the cube:

Calculate the area of each face.

Since it is a cube, all of the faces are the squares and of equal size.

Area of a square formula: $\text{(side length)}^2$

So, the area of each face $=9 \times 9=81cm^2$

Add the areas together.

There are $6$ faces, each with an area of $81cm^2$

$6 \times 81=486$

Write the answer, including the units.

The measurements of the cube are in $cm$ so the area will be measured in $cm^2$

Total surface area $= 486cm^2$

Example 3: surface area of a triangular prism

Work out the surface area of the triangular prism

Calculate the area of each face.

Face	Area (cm²)
Front	½ × 12 × 8 =48
Back	48
Bottom	12 × 7 = 84
Left side	10 × 7 = 70
Right side	70

Add the areas together.

$48+48+84+70+70$

Write the answer, including the units.

The units are in $mm$ so the area will be measured in $mm^2$

Surface area $= 320mm^2$

Example 4: surface area of a cylinder

The radius of this cylinder is $5m$ and the height of the cylinder is $2m.$ Work out the surface area of the cylinder. Give your answer to $3$ significant figures.

Calculate the area of each face.

As the area of each face includes the value of $\pi$ , leave the area of each face in terms of $\pi$ , then we can add them together more easily.

$\begin{aligned} \text{Curved surface area (lateral surface area)}&=2 \pi rh\\\\ &=2 \times \pi \times 5 \times 2\\\\ &=20\pi \end{aligned}$

The cross section is a circle.

$\begin{aligned} \text{Area of circle }&=\pi r^{2}\\\\ &=\pi \times 5^{2}\\\\ &=25\pi \end{aligned}$

The base is the same as the top so the area of the base is also $25\pi$ .

Add the areas together.

$20\pi+25\pi+25\pi=70\pi$

$70\pi=219.9114858...$

Write the answer, including the units.

We need to round the answer to $3$ significant figures.

Surface area $= 220m^2 \; (3sf)$

Example 5: surface area of a cone

Work out the surface area of the cone. Give your answer to $1$ decimal place.

Calculate the area of each face.

Similar to the cylinder, the area of each face of the cone is calculated using $\pi$ . To avoid any rounding errors, keep the answer in terms of $\pi$ , then round the answer at the very end of the calculation.

$\begin{aligned} \text{Curved surface area }&=\pi rl\\\\ &=\pi \times 5 \times 13\\\\ &=65\pi \end{aligned}$

Notice that we use the slant height of the cone, not the vertical height.

The base is a circle so the area of the base $= \pi r^2$

$\begin{aligned} \text{Area of a circle }&=\pi r^{2}\\\\ &=\pi \times 5^{2}\\\\ &=25\pi \end{aligned}$

Add the areas together.

$65\pi+25\pi=90\pi$

$90\pi=282.7433388...$

Write the answer, including the units.

We need to write the answer to $1$ decimal place.

Total surface area $= 282.7cm^2 \; (1dp).$

Example 6: surface area of a sphere

Work out the surface area of the sphere. Give your answer to the nearest integer.

Calculate the area of each face.

The surface area formula for a sphere is:

$\begin{aligned} \text{Surface area }&=4 \pi r^{2}\\\\ &=4 \times \pi \times 16^{2}\\\\ &=3216.9908… \end{aligned}$

Add the areas together.

A sphere only has one face.

Write the answer, including the units.

We need to write our answer to the nearest integer.

Surface area $= 3217cm^2$

Common misconceptions

Calculating with different units

You need to make sure all measurements are in the same units before calculating surface area. (For example you can’t have some measurements in $cm$ and some in $m$ ).

Make sure you have the correct units

For area we use square units such as $cm^2$ (square centimetres), $m^2$ (square metres), $in^2$ (square inches) and $ft^2$ (square feet),

For volume we use cube units such as $cm3, \; m^3, \; km^3$

Calculating volume instead of surface area

Volume and surface area are different things – volume tells us the space within the shape whereas surface area is the total area of the faces. To find surface area, work out the area of each face and add them together.

Rounding

It is important to not round decimals until the end of the calculation. Rounding too early will result in an inaccurate answer. Keep as many values in terms of $\pi$ until you need to calculate the final rounded answer.

Practice surface area questions

$92cm^2$

$48cm^2$

$108cm^2$

$76cm^2$

Calculating the area of each face, we have:

Face	Area (cm²)
Front	8 × 2 =16
Back	16
Bottom	8 × 3 = 24
Top	24
Left side	2 × 3 = 6
Right side	6

Total surface area: $16+16+24+24+6+6=92cm^2$

$360cm^2$

$240cm^2$

$300cm^2$

$480cm^2$

Calculating the area of each face, we have:

Face	Area (cm²)
Front	½ × 5 × 12 =30
Back	30
Bottom	8 × 12 = 96
Top	8 × 13 = 104
Left side	8 × 5 = 40

Total surface area $= 30+30+96+104+40=300cm^2$

$464mm^2$

$432mm^2$

$400mm^2$

$440mm^2$

Calculating the area of each face, we have:

Face	Area (cm²)
Front	½(2 + 8) × 4 =20
Back	20
Bottom	8 × 20 = 160
Top	2 × 20 = 40
Left side	5 × 20 = 100
Right side	100

Total surface area $= 20+20+160+40+100+100=440mm^2$

$1190m^2$

$633m^2$

$449m^2$

$346m^2$

$\begin{aligned} \text{Curved surface area }&=2 \pi rh\\\\ &=2 \times \pi \times 6.5 \times 9\\\\ &=117\pi \end{aligned}$

$\begin{aligned} \text{Area of a circle }&=\pi r^{2}\\\\ &=\pi \times 6.5^{2}\\\\ &=\frac{169}{4}\pi \end{aligned}$

Total surface area: $117\pi+\frac{169}{4}\pi+\frac{169}{4}\pi=\frac{403}{2}\pi$

$\frac{403}{2}\pi=633.0309197…$

Surface area $= 633m^2 \; (3sf)$

$29.5cm^2$

$22.6cm^2$

$7.5cm^2$

$23.9cm^2$

$\begin{aligned} \text{Curved surface area }&=\pi rl\\\\ &=\pi \times 2 \times 2.7\\\\ &=5.4\pi \end{aligned}$

$\begin{aligned} \text{Area of a circle }&=\pi r^{2}\\\\ &=\pi \times 2^{2}\\\\ &=4\pi \end{aligned}$

Total surface area: $=5.4\pi+4\pi=9.4\pi$

Surface area $= 29.5cm^2 \; (1dp)$

$201.06m^2$

$2.01m^2$

$0.27m^2$

$25.27m^2$

$\begin{aligned} \text{Surface area of a sphere }&=4 \pi r^{2}\\\\ &=4 \times \pi \times 0.4^{2}\\\\ &=0.64\pi \\\\ &=2.010619298… \end{aligned}$

Surface area $= 2.01m^2 \; (2dp)$

Surface area GCSE questions

1. Calculate the surface area of the cone. Give your answer to the nearest integer.

How to calculate surface area GCSE Question 1

(4 marks)

Show answer

$\begin{aligned} \text{Curved surface area }&=\pi \times 21 \times 37\\\\ &=2441.017492…\text{ or }777\pi \end{aligned}$

(1)

$\begin{aligned} \text{Area of circle }&=\pi \times 21^{2}\\\\ &=1385.4423…\text{ or }441\pi \end{aligned}$

(1)

$\text{Total surface area } = \text{their }777\pi+ \text{their } 441\pi =1218\pi =3826.459852…$

(1)

$\text{Surface area } = 3826cm^2$

(1)

2. A drinks company is designing a new product. Can $A$ has a volume of $108\pi \text{ cm}^3$ and Can $B$ has a volume of $98\pi \text{ cm}^3.$

How to calculate surface area GCSE Question 2 Image 1

The company wants to minimise their packaging for the new product. Which should they choose? Explain your answer.

(9 marks)

Show answer

Can $A$

$\begin{aligned} \text{Curved surface area }&=2 \times \pi\times 3 \times 12\\\\ &=226.1946711…\text{ or }72\pi \end{aligned}$

(1)

$\begin{aligned} \text{Area of circle }&=\pi \times 3^{2}\\\\ &=28.27433388…\text{ or }9\pi \end{aligned}$

(1)

$\text{Total surface area: }\text{their }72\pi+(2\times\text{their }9\pi)=90\pi=282.7433388…$

(1)

Can $B$ :
$\begin{aligned} \text{Curved surface area }&=2 \times \pi\times 3.5 \times 8\\\\ &=175.9291886…\text{ or }56\pi \end{aligned}$

(1)

$\begin{aligned} \text{Area of circle }&=\pi \times 3.5^{2}\\\\ &=38.48451001…\text{ or }12.25\pi \end{aligned}$

(1)

$\text{Total surface area: }\text{their }56\pi+(2\times\text{their }12.25\pi)=80.5\pi=252.8982086…$

(1)

Volume to Surface Area ratio for Can $A$

$\begin{aligned} &108 \pi :90 \pi \\\\ &=1:0.83 \end{aligned}$

(1)

Volume to Surface Area for Can $B$

$\begin{aligned} &98 \pi :80.5 \pi \\\\ &=1:0.82 \end{aligned}$

(1)

They should use can $B$ because it has a smaller volume to surface area ratio.

(1)

3. The surface area of this sphere is $200cm^2.$ Calculate the radius of the sphere. Give your answer to $3sf.$

How to calculate surface area GCSE Question 3

(4 marks)

Show answer

$4\pi r^{2}=200$

(1)

$r^{2}=\frac{200}{4\pi}=15.91549431…$

(1)

$r=\sqrt{15.91549431…}=3.989422804…$

(1)

$r=4.00cm \; (3sf)$

(1)

Learning checklist

You have now learned how to:

Find the surface area of cuboids and prisms
Find the surface area of cylinders, cones and spheres
Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3D

The next lessons are

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Introduction

What is surface area?

How to calculate surface area

Surface area worksheet

Surface area examples

Example 1: surface area of a cuboid Example 2: surface area of a cube Example 3: surface area of a triangular prism Example 4: surface area of a cylinder Example 5: surface area of a cone Example 6: surface area of a sphere

Common misconceptions

Practice surface area questions

Surface area GCSE questions

Learning checklist

Next lessons

Still stuck?