GCSE Maths Geometry and Measure Cuboid

Surface Area of a Cuboid

Surface Area of a Cuboid

Here we will learn about the surface area of a cuboid and how to calculate it.

There are also volume and surface area of a cuboid 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 cuboid?

The surface area of a cuboid is the total area of all of the faces of a cuboid (or rectangular prism).

The 3 dimensions of a cuboid are width, length and height.

Cuboids have three pairs of identical faces – top and bottom, front and back, and left and right.

To work out the total surface area of a cuboid, we need to work out the area of each rectangular face and add them all together.

E.g. Find the surface area of a cuboid.

Since it is an area, surface area is measured in square units (e.g. mm^2, cm^2, m^2 etc).

What is the surface area of a cuboid?

What is the surface area of a cuboid?

How to calculate the surface area of a cuboid

In order to work out the surface area of a cuboid:

  1. Work out the area of each face.
  2. Add the six areas together.
  3. Include the units.

How to calculate the surface area of a cuboid

How to calculate the surface area of a cuboid

Surface area of a cuboid worksheet

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

COMING SOON
x

Surface area of a cuboid worksheet

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

COMING SOON

Surface area of a cuboid examples

Example 1: surface area of a cuboid

Work out the surface area of the cuboid

  1. Work out the area of each face.

The area of the bottom is 9\times3=27cm^2 .

The top face is the same as the bottom face so the area of the top is also 27cm^2 .

The area of the front is 9\times4=36cm^2 .

The back face is the same as the front face so the area of the back is also 36cm^2 .

The area of the right hand side is 3\times4=12cm^2 .

The left side face is the same as the right side face so the area of the left side is also 12cm^2 .

It will make our working clearer if we use a table:

FaceArea
Bottom9×3=27
Top27
Front9×4=36
Back36
Right side3×4=12
Left side 12

2Add the six areas together.

The sum of the areas is: 27+27+36+36+12+12=150

3Include the units.

The measurements on the cuboid are in cm therefore the total surface area of the cuboid = 150cm^2 .

Example 2: surface area of a cuboid

Work out the surface area of the cuboid

FaceArea
Bottom6×3=18
Top18
Front9×6=54
Back54
Right side9×3=27
Left side 27

18+18+54+54+27+27=198

The measurements on the cuboid are in mm therefore the total surface area of the cuboid = 198mm^2 .

Example 3: surface area of a cube

Work out the surface area of this cube

Each face of a cube is the same. For this cube, the area of each face is 8\times8=64cm^2

6\times64=384

The measurements on the cube are in cm therefore the total surface area of the cube = 384cm^2 .

Example 4: surface area with different units

Work out the surface area of this cuboid

Notice that one of the measurements is in metres and the rest are in centimetres. Before we can calculate any areas, we need to ensure all units are the same. In this case, 0.1m=10cm so we can use 10cm .

FaceArea
Bottom8×10=80
Top80
Front8×2.5=20
Back20
Right side10×2.5=25
Left side 25

80+80+20+20+25+25=250

The measurements we have used are in cm therefore the total surface area = 250cm^2 .

Example 5: surface area using algebra

Work out the surface area of this cuboid.

FaceArea
Bottom14×x=14x
Top14x
Front14×6=84
Back84
Right side6×x=6x
Left side 6x

14x+14x+84+84+6x+6x = 40x+168

The measurements we have used are in cm therefore the total surface area = (40x+168) cm^2 .

If we are told the value of the surface area, we can use the expression we have found to work out the value of x .

Let’s say the surface area of this cuboid is 328cm^2 . Then we can say:

40x+168=328 .

Now we can solve this equation:

40x+168=328

40x=160

x=4cm

The width of this cuboid is 4cm .

Example 6: surface area using algebra

Work out the surface area of this cuboid

FaceArea
Bottom10×4y=40y
Top40y
Front10×2y=20y
Back20y
Right side2y×4y=8y2
Left side 8y2

40y+40y+20y+20y+8y^2+8y^2= 16y^2+120y

The measurements we have used are in m therefore the total surface area = (16y^2+120y) m^2 .

Common misconceptions

  • Calculating volume instead of surface area

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

  • Equal faces

A common mistake is to think that four of the faces are equal.

E.g.
The first pair of faces are equal to each other.
The second pair of faces are equal to each other.
The third pair of faces are equal to each other.

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

1. Work out the surface area of the cuboid

 

 

315\mathrm{cm}^{2}
GCSE Quiz False

222\mathrm{cm}^{2}
GCSE Quiz False

342\mathrm{cm}^{2}
GCSE Quiz True

462\mathrm{cm}^{2}
GCSE Quiz False

Work out the area of each of the six faces:

Face Area
Bottom 15=105
Top 105
Front 15=45
Back 45
Right side 7=21
Left side  21

 

\text{Total surface area: }105+105+45+45+21+21=342\mathrm{cm}^{2}

2. Work out the surface area of this cube

 

96\mathrm{cm}^{2}
GCSE Quiz True

64\mathrm{cm}^{2}
GCSE Quiz False

24\mathrm{cm}^{2}
GCSE Quiz False

256\mathrm{cm}^{2}
GCSE Quiz False

Since it is a cube, all of the faces are the same. The area of each face is 4\times 4=16\mathrm{cm}^{2} . There are six identical faces therefore the total surface area of the cube is 6 \times 16=96 \mathrm{cm}^{2}

3. Work out the surface area of this cuboid

 

 

 

33.56\mathrm{cm}^{2}
GCSE Quiz False

4.2\mathrm{cm}^{2}
GCSE Quiz False

8900\mathrm{cm}^{2}
GCSE Quiz True

12400\mathrm{cm}^{2}
GCSE Quiz False

Some of the measurements are in m and one is in cm. We need all of the measurements to be in the same units so convert the metres to centimetres. 0.7m=70cm and 0.4m=40cm . Now we can calculate the areas:

 

Face Area
Bottom 70×15=1050
Top 1050
Front 70×40=2800
Back 2800
Right side 40×15=600
Left side  600

 

\text{Total surface area: } 1050+1050+2800+2800+600+600=8900\mathrm{cm}^{2}

4. Work out the surface area of this cuboid

 

 

16a \mathrm{cm}^{2}
GCSE Quiz False

15a \mathrm{cm}^{2}
GCSE Quiz False

(20a+30) \mathrm{cm}^{2}
GCSE Quiz False

(16a+30) \mathrm{cm}^{2}
GCSE Quiz True

Work out the area of each of the six faces:

 

Face Area
Bottom a=5a
Top 5a
Front a=3a
Back 3a
Right side 5=15
Left side  15

 

\text{Total surface area: } 5a+5a+3a+3a+15+15=(16a+30) \mathrm{cm}^{2}

5. Work out the surface area of the cuboid

 

 

80b \mathrm{m}^{2}
GCSE Quiz False

16b^{2} \mathrm{m}^{2}
GCSE Quiz False

(4b^{2}+48b) \mathrm{m}^{2}
GCSE Quiz True

(8b^{2}+16b) \mathrm{m}^{2}
GCSE Quiz False

Work out the area of each of the six faces:

 

Face Area
Bottom b=8b
Top 8b
Front 2b=2b2
Back 2b2
Right side 2b=16b
Left side  16b

 

\text{Total surface area: }8b+8b+2b^{2}+2b^{2}+16b+16b=(4b^{2}+48b)m^{2}

6. Given that the surface area of this cuboid is 142\mathrm{cm}^{2} find the value of x .

 

 

6.76cm
GCSE Quiz False

5cm
GCSE Quiz True

14.2cm
GCSE Quiz False

20.29cm
GCSE Quiz False

Work out the area of each of the six faces:

 

Face Area
Bottom x=7x
Top 7x
Front x=3x
Back 3x
Right side 7=21
Left side  21

 

\text{Total surface area: }7x+7x+3x+3x+21+21=20x+42

 

Since we know the surface area is  142\mathrm{cm}^{2}
we can say:
20x+42=142\\ 20x=100\\ x=5\mathrm{cm}

Surface area of a cuboid GCSE questions

1. Calculate the surface area of the cuboid.

 

 

(3 marks)

Show answer

Two of:
12\times 3.5=42\\ 12\times 4=48\\ 4\times 3.5=14

(1)

42+42+48+48+14+14

(1)

208\mathrm{cm}^{2}

(1)

2. A breakfast cereal producer wants to produce a cereal box with a volume of 4800cm^3 . The company wants to use as little cardboard as possible for each box. Should they use box A or box B ? You must show your working.

 

     

 

(5 marks)

Show answer

Box A surface area: 160+160+600+600+240+240

(1)

\text{Surface area }=2000\mathrm{cm}^{2}

(1)

Box B surface area: 192+192+480+480+250+250

(1)

\text{Surface area }=1844\mathrm{cm}^{2}

(1)

They should use box B

(1)

3. John wants to paint 4 identical doors, as shown below.

 

1 litre of paint will cover
10\mathrm{m}^{2}
John has 1 litre of paint. Does he have enough paint to cover all 4 doors? You must show your working.

 

(5 marks)

Show answer
5cm = 0.05m

(1)

Surface area of 1 door: 0.03+0.03+1.2+1.2+0.1+0.1

(1)

\text{Surface area of 1 door }=2.66\mathrm{m}^{2}

(1)

\text{Surface area of 4 doors: }4 \times 2.66=10.64\mathrm{m}^{2}

(1)

No he does not have enough paint

(1)

Learning checklist

You have now learned how to:

  • Calculate the surface area of a cuboid
  • Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3D

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.

Find out more about our GCSE maths revision programme.