GCSE Maths Geometry and Measure 3D Shapes Cuboid
Surface Area
Here we will learn about the surface area of a cube, including how to calculate the surface area of a cube and how to find missing values of a cube given its surface area.
There are also surface area of a cube 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 cube?
The surface area of a cube is the sum of the areas of all the faces of a cube. A cube is a three-dimensional solid object that has six congruent square faces. This means they are all the same size.
To find the area of each face we multiply the side lengths together. We then multiply the area of each of the square faces by six.
The formula to calculate the surface area, S, of a cube is
S=6x^{2}
where x represents the side length of the cube.
We can use this formula to find the surface area of any cube.
Surface area is measured in square units, for example mm^{2}, \ cm^{2} or m^{2}.
What is the surface area of a cube?
How to calculate the surface area of a cube
In order to calculate the surface area of a cube:
- Write the formula for the surface area of the cube.
- Substitute any known value(s) into the formula.
- Complete the calculation.
- Write the solution, including the units.
Explain how to calculate the surface area of a cube
Volume and surface area of a cube worksheet
Get your free surface area of a cube worksheet of 20+ volume and surface area of a cube questions and answers. Includes reasoning and applied questions.
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Volume and surface area of a cube worksheet
Get your free surface area of a cube worksheet of 20+ volume and surface area of a cube questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREE
Surface area of a cube examples
Example 1: integer side lengths
Work out the surface area of the cube below.
- Write the formula for the surface area of the cube.
S=6x^{2}
2Substitute any known value(s) into the formula.
Here, x=5 and so we have
S=6\times{5}^{2}.
3Complete the calculation.
S=6\times{5}^{2}=6\times{25}=150
4Write the solution, including the units.
As the unit of length is centimetres (cm), the unit of area is square centimetres (cm^{2}).
S=150cm^{2}.
Example 2: one known edge of the cube
Work out the surface area of the cube.
Write the formula for the surface area of the cube.
S=6x^{2}
Substitute any known value(s) into the formula.
Substituting x=6 into the formula, we have
S=6\times{6}^{2}.
Complete the calculation.
S=6\times{6}^{2}=6\times{36}=216
Write the solution, including the units.
As the unit of length is centimetres (cm), the unit of area is square centimetres (cm^{2}).
S=216cm^{2}
Example 3: worded problem
A cube structure has a side length of 7m. Calculate the total surface area of the structure.
Write the formula for the surface area of the cube.
S=6x^{2}
Substitute any known value(s) into the formula.
Substituting x=7 into the formula, we have
S=6\times{7}^{2}.
Complete the calculation.
S=6\times{7}^{2}=6\times{49}=294
Write the solution, including the units.
As the unit of length is metres (m), the unit of area is square metres (m^{2}).
S=294m^{2}
Example 4: area of a face given
The area of the face of a cube is 30cm^{2}. Work out the surface area of the cube.
Write the formula for the surface area of the cube.
S=6x^{2}
Substitute any known value(s) into the formula.
As we know the area of one face of the cube, we can express this as x^{2}=30 as x is the side length of the cube, and we know the area, x^{2}.
Substituting x^{2}=30 into the formula, we have
S=6\times{30}.
Complete the calculation.
S=6\times{30}=180
Write the solution, including the units.
As the unit of area is square centimetres (cm^{2}), we can use this in the solution.
S=180cm^{2}
Example 5: find the length of a cube given the surface area
The surface area of a cube is 24cm^{2}. Work out the length of the cube.
Write the formula for the surface area of the cube.
S=6x^{2}
Substitute any known value(s) into the formula.
Here we know that S=24 and so substituting this into the formula, we have
24=6\times{x}^{2}.
Complete the calculation.
Dividing both sides by 6, we have
\begin{aligned}24\div{6}&=x^{2}\\\\4&=x^{2} \\\\x&=2\end{aligned}
Write the solution, including the units.
As the unit of area is square centimetres (cm^{2}), the unit length will be in centimetres.
x=2cm
Example 6: find the length of a cube given the surface area (decimal solution)
The surface area of a cube is 483mm^{2}. Work out the length of the side x correct to 2 decimal places.
Write the formula for the surface area of the cube.
S=6x^{2}
Substitute any known value(s) into the formula.
As we know the surface area, we can substitute S=483 into the formula
483=6\times{x}^{2}.
Complete the calculation.
To complete the calculation, we need to divide both sides by 6 first, and then square root both sides to find x.
\begin{aligned}483\div{6}&=x^{2}\\\\80.5&=x^{2}\\\\\sqrt{80.5}&=x\\\\x&=8.972179222463…\end{aligned}
Write the solution, including the units.
x=8.97mm \ (2dp).
Common misconceptions
- Missing/incorrect units
You should always include units in your answer.
Surface area is measured in square units (for example, mm^2, \ cm^2, \ m^2 etc.).
- Calculating with different units
You need to make sure all measurements are in the same units before calculating the surface area.
For example, you can’t have some in cm and some in m.
- Calculating volume instead of surface area
Volume and surface area are different quantities. The volume of the cube is the three-dimensional space in a shape and is measured in cubic units.
Surface area is the amount of space covering the outside of a 3D shape. To find surface area, we need to work out the area of one face and multiply it by six.
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:
- Cuboid
- Volume of a cuboid
- Surface area of a cuboid
- Volume of a cube
Practice surface area of a cube questions
1. Work out the surface area of the cube.
27cm^3
12cm^2
36cm^2
54cm^2
S=6x^{2} where x=3cm.
S=6\times{3}^{2}=6\times{9}=54cm^{2}.
2. Calculate the surface area of the cube below. Write your answer in square centimetres.
0.125cm^3
15,000cm^2
6cm^2
1.5cm^2
S=6x^{2} where x=0.5m = 50cm.
S=6\times{50}^{2}=6\times{2500}=15,000cm^{2}.
3. Work out the surface area of the cube. Give your answer in cm^{2}.
96cm^2
9,600cm^2
960cm^2
9.6cm^2
S=6x^{2} where x=40cm=0.4m.
S=6\times{40}^{2}=6\times{1600}=9,600cm^{2}.
4. The surface area of a cube is 150cm^{2}. Find the length of the side of the cube.
135 \ 000cm
2.04cm
5cm
12.5cm
S=6x^{2} where S=150cm^{2}.
\begin{aligned}150&=6\times{x}^{2}\\\\25&=x^{2}\\\\x&=\sqrt{25}\\\\x&=5\end{aligned}
5. The surface area of a cube is 6m^{2}. Work out the length of each side x.
2m
1m
0.3m
0.41m
S=6x^{2} where S=6m^{2}.
\begin{aligned}6&=6\times{x}^{2}\\\\6\div{6}&=x^{2}\\\\x^{2}&=1\\\\x&=\sqrt{1}\\\\x&=1\end{aligned}
6. The surface area of a cube is 186m^{2}. Work out the length of each side. Write your answer to the nearest centimetre.
5.57m
2.27m
15.50m
207,576m
S=6x^{2} where S=186m^{2}.
\begin{aligned}186&=6\times{x}^{2}\\\\31&=x^{2} \\\\x&=\sqrt{31}=5.567764362830…=5.57\text{ (2dp)}\end{aligned}
Surface area of a cube GCSE questions
1. Here is a cube.
The cube has a volume of 216cm^{3}.
Given that V=x^{3}, where x is the side length of the cube, work out the total surface area of the cube.
(4 marks)
Show answer
x=\sqrt[3]{216} \ (=6cm)
(1)
6 \times 6 \ or \ 36cm^2
(1)
6 \times 6 \times 6 \ or \ 36 \times 6
(1)
216cm^2
(1)
2. The diagram shows a cube of side 3cm.
Determine the volume : surface area ratio of the cube.
Write your answer as a ratio in the simplest form.
(5 marks)
Show answer
3 \times 3 \times 6 \ or \ 9 \times 6
(1)
54cm^2
(1)
3 \times 3 \times 3 = 27cm^3
(1)
27:54
(1)
1:2
(1)
3. The total surface area of a cube is 294cm^{2}. Work out the side length of the cube.
(3 marks)
Show answer
294 \div{6} \ or \ 49
(1)
\sqrt{49}
(1)
7cm
(1)
Learning checklist
You have now learned how to:
- Calculate the surface area of a cube
- Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3D
The next lessons are
- Prism shape
- Triangular prism
- Sphere
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