# 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.

## 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:*

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

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

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