Nets and Surface Area

A net is a deconstructed 3D shape folded out flat. Nets can be helpful when we want to calculate the surface area of a 3D shape.

Skill 1: Nets of Cubes

A net shows us each face of a shape when it is laid out flat, and there are often many different nets for a 3D shape. Below are some examples of nets of cubes.

Note: these are only some of the nets of a cube – there are many more

Each of the nets above can be folded up to construct a cube, like so:

Cubes are the simplest nets you will encounter, so make sure you are comfortable drawing them before moving on to harder shapes.

Skill 2: Nets of Other Shapes

There are some other common shapes that you should familiarise yourself with, including prisms and pyramids.

A prism is like a 2D shape ‘stretched out’ with the original 2D at either end. This makes drawing a net of a prism fairly simple, as it will feature the 2D shape at both ends with rectangles between them that form the ‘stretched out’ section. A cuboid is also a type of prism, so the net of a cuboid follows the same pattern.

A pyramid always features a 2D shape as its base, with each edge linked to a triangular face. These triangular faces all converge to a central point above the centre of the base. The nets of pyramids are easy to draw – just draw the shape of the base with triangles attached to each side.

A cylinder can be thought of as a circular prism. Its net follows the same pattern as the nets of prisms.

Some examples of nets of 3D shapes are shown below:

Skill 3: Finding the Surface Area

We can use nets to find the surface area of a 3D shape – the combined area of all the faces.

Consider the following net of a cuboid.

We can use the net to work out the surface area. There are $3$ ‘pairs’ of faces that are the same – if we calculate the areas of these individual faces we can add them all up to get the surface area of the cuboid.

The total surface area is therefore:

$2 times48 + 2 times32 + 2 times 24 = 208$ cm $^2$

Note: remember that the units of area are cm $^2$ , m $^2$ etc.

Example: Surface Area of a Cylinder

Calculate the surface area of cylinder $A$ from the net shown.

[3 marks]

The diameter of the circular face is $6$ cm, meaning the radius of the circle is $3$ cm. We can calculate the area of one circular face using the formula:

Area of a circle $= pi r^2$

So the area of both circular faces on the cylinder is:

$2times pi times 3^2=18pi = 56.55$ cm $^2$

To calculate the area of the rectangle, we need to know the length – this is the same as the circumference of the circle ( $2pi r$ ).

The length of the rectangle is therefore:

$2pi r = 6 pi$ cm

We can now calculate the area of the rectangle face:

$6 pi times 6 = 36 pi = 113.10$ cm $^2$

The total surface area of the cylinder is therefore:

$text{Area} = 56.55 + 113.10=169.65$ cm $^2$

Note:

There is a simple formula for calculating the surface area of a cylinder:

Surface area of a cylinder $=2pi rh + 2pi r^2$

Where $r$ is the radius of the circle face and $h$ is the height of the cylinder.

Substituting values into this formula is a quick way of performing the calculations shown in the example above.

Nets and Surface Area Example Questions

Question 1: State which of the following nets can be used to form a cube.

[2 marks]

Only nets A and C will form a cube.

The faces on net B will overlap when folded over and net D has too many faces.

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Question 2: Sketch a net of the following cuboid and label all the measurements.

cuboid net question

[3 marks]

Your completed sketch should look something like this:

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Question 3: Calculate the surface area of the triangular prism from the net shown.

triangular prism surface area question

We need to use the following formula:

Area of a triangle $=dfrac{1}{2} times text{base} times text{height}$

So the area of both triangular faces is:

$2times dfrac{1}{2}times4times3 = 12$ cm $^2$

The area of the three rectangular faces is:

$3times10times4 = 120$ cm $^2$

The total surface area is:

$12+120=132$ cm $^2$

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Question 4: Calculate the surface area of the cylinder from the net shown.

cylinder net question

Use the formula and substitute in the values of $r=2$ and $h= 8$ :

$begin{aligned}text{Surface area} &= 2pi r^2 + 2pi rh &=(2 times pi times 2^2) + (2times pi times 2 times 8) &= 8pi + 32pi &= 25.13+100.53 &=125.66 text{ cm}^2end{aligned}$