Nets and surface area

​​In a nutshell

Finding the surface area of a 3D shape is much easier to approach by using nets. A net of a 3D shape is a 2D representation of the faces of the 3D shape.

Nets

Definition

A net is a 3D shape that has been unfolded and laid out flat. There is often more than one net for a particular shape. Here are some examples of nets.

​​SHAPE	NET
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Finding surface area using nets

Finding the surface area of a 3D shape is the same as finding the total area of the net of the shape. A net is usually a compound shape, so you can find the area of the net by working out the areas of the individual shapes.

Example 1

A cuboid has base, width and height to be $$4cm, 7cm$$ and $$5cm$$ respectively. Work out the surface area of the cuboid.

First, sketch the net of the cuboid. The cuboid is made up of six rectangles that come in three pairs, so the net will look like this:

Now, work out the area of the net:

The net has two rectangles with side lengths $$4cm$$ and $$7cm$$. So, their combined area is $$2\times4\times7=56cm^2$$.

The net has two rectangles with side lengths $$7cm$$ and $$5cm$$. So, their combined area is $$2\times7\times5=70cm^2$$.

The net has two rectangles with side lengths $$4cm$$ and $$5cm$$. So, their combined area is $$2\times4\times5=40cm^2$$.

Then, add up all the individual areas to give the total area of the net - and hence the surface area:

​$$56+70+40=166$$​​

The surface area of the cuboid is $$\underline{ 166 \ cm^2}$$.

Cylinders

The net of a cylinder consists of two circles and a rectangle.

While it is possible to use the net to work out the surface area, it is easier to just memorise the formula in this case.

​$$\text{Surface area of a cylinder}=2\pi r^2+2\pi rh$$​

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

Example 2

What is the exact surface area of a cylinder with a radius of $$7cm$$ and a height of $$1cm$$?

Substitute $$r=7,h=1$$ into the formula:

$$\begin{aligned}\text{Surface area of a cylinder}&=2\pi r^2+2\pi rh\\&=2\pi (7)^2+2\pi (7)(1)\\&=2\pi (49) + 2\pi (7)\\&= 98\pi + 14\pi\\&=112\pi \end{aligned}$$​​

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The question is asking for the exact surface area, so leave the answer in terms of $$\pi$$.

The surface area of the cylinder is $$\underline{112\pi \ cm^2}$$.