Surface Area (VCE SSCE General Mathematics): Revision Notes

Surface Area

What is surface area?

Surface area is the total area of all the outer surfaces of a three-dimensional object. To calculate it, you need to find the area of each individual surface and then add them all together.

The surface area is always measured in square units, such as $\text{cm}^2$, $\text{m}^2$, or $\text{mm}^2$.

Solids with plane faces

Many solid objects, including prisms and pyramids, have flat surfaces called plane faces. When working with these shapes, it's helpful to use a net to ensure you've counted all the surfaces.

What is a net?

A net is a flat, two-dimensional pattern that shows all the faces of a solid shape. When you fold the net along its edges, it forms the complete three-dimensional object.

For example, here are the nets of a cube and a square pyramid:

Finding the surface area of a pyramid

Let's work through an example to see how nets help us calculate surface area.

Solids with curved surfaces

Some three-dimensional objects have curved surfaces rather than flat faces. The most common shapes with curved surfaces are cylinders, cones, and spheres. For these shapes, we use special formulas to calculate surface area.

Surface area formulas

Here are the key formulas you need to know:

Cylinder:

$SA = 2\pi r^2 + 2\pi rh$

where $r$ is the radius and $h$ is the height.

Sphere:

$SA = 4\pi r^2$

where $r$ is the radius.

Cone (for reference only):

$SA = \pi r^2 + \pi rs$

where $r$ is the radius and $s$ is the slant height.

Understanding the cylinder formula

To understand where the cylinder formula comes from, we can draw its net:

Cylinder example

Sphere example

Exam tips

Remember!