Surface area of 3D shapes: Nets, formulae

In a nutshell

The surface area of a shape is the sum of the areas of its faces. The surface area of 3D shapes can either be calculated using nets or using given formulae.

Surface area using nets

Net

A net is a 3D shape that has been unfolded and laid out flat. Here are some examples of nets.

SHAPE	NET

Surface area

To find the surface area of a shape using nets, sketch the net and work out the area of the net. This can be done by working out the area of each individual shape in the net.

Example 1

What is the surface area of a cube with side lengths $8cm$ ?

First, sketch the net:

Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae

The net of a cube is six squares. In this example, each square will have side lengths $8cm$ .

Then, find the area of the net:

The net is a compound shape consisting of six squares.

The area of one of these squares is given to be:

$8\times8=64cm^2$ 

The area of the net is therefore:

$6\times64=384cm^2$

The surface area of the cube is $\underline{384 \ cm^2}$ .

Surface area formulae

These are the formulae you need to know:

SHAPE	FORMULA	DIAGRAM
Sphere	$S.A=4\pi r^2$
Cylinder	$S.A=2\pi r^2 +2\pi rh$
Cone	$\text{Curved surface area}=\pi rl$ $S.A=\pi rl +\pi r^2$

Example 2

What is the surface area of a sphere with radius $2m$ to three significant figures?

Substitute $r=2$ into the formula:

$\begin{aligned}S.A&=4\pi r^2\\&=4\pi (2)^2\\&=16\pi\\&=50.26548246...m^2\end{aligned}$

The surface area is $\underline{50.3m^2\space\ (3\space s.f.)}$ .