Volume is a measure of the space inside a 3D (solid shape). It is measured using units such as cubic centimetres ($cm^3$cm3) and cubic metres ($m^3$m3) in the metric system and cubic feet ($ft^3$ft3) in the imperial system.

For example, a cubic centimetre is a cube that is $1$1 cm long, $1$1 cm wide and $1$1 cm high like the one below:

So when we find the volume of a solid shape, we are basically working out how many little cubes would fit inside the whole space.

There are lots of formulas for calculating volume but generally for prisms:

Volume of a prism = Area of the cross section x Height

Unfortunately, we don't always have little cubes or even pictures to help us calculate it. However, the great news is that we can work out the volume of different solids in one step without any of this! We are going to start by looking at how to calculate the volume of cubes and rectangular prisms.

For example, to make a $10$10 rod like the one in the picture below, you would need $10$10 little cubic centimetre blocks. In other words, the volume of the rod would be $10$10cm³.

Rather than count the number of cubes used to make the shape, we could use the dimensions to calculate the volume. The rod is $10$10cm long, $1$1cm wide and $1$1cm tall.

$Volume$`Volume`	$=$=	$length\times width\times height$`length`×`width`×`height`
	$=$=	$10\times1\times1$10×1×1
	$=$=	$10$10$cm^3$`cm`3

You can see we get the same answer, $10$10cm³.

To calculate the volume of cubes or rectangular prisms

$\text{Volume }$Volume $=$= $\text{length }$length $\times$× $\text{width }$width $\times$× $\text{height }$height

How do we know this? Well look at the table below.

	Measured with	Number of dimensions	Dimensions
Straight line	length	one	length
Square	area	two	length x width
Cube	volume	three	length x width x height

Have a play with the applet below to see how the volume of a shape can change and practice calculating what the volume would be.

Created with Geogebra

Capacity

Capacity is often confused with volume but it's a little bit different. Capacity is the amount a 3D object can hold. It is measured in units such as millilitres, litres or kilolitres in the metric system and pints or gallons in the imperial system.

For example, this carton of orange juice has a capacity of $1$1 litre.

Converting between Volume and Capacity

Here is a table of conversions between volume and capacity. Take your time learning these and they are important to remember.

Volume	Capacity
$1$1 cm³	$1$1 mL
$1000$1000 cm³	$1$1 L
$1$1 m³	$1000$1000 L
$1$1 m³	$1$1 kL

Worked Examples

Question 1

If a solid figure has a volume of $33$33 cm³, how many unit cubes would fit inside the solid?

Question 2

Convert $40$40 millilitres (mL) to cubic centimetres (cm³).

Question 3

A container has the shape of a rectangular prism, with the following dimensions: $30$30 cm, $80$80 cm, and $10$10 cm.

What is the volume of this container?
Find the capacity of the container in litres.

Outcomes

5.M2.07

Determine, through investigation, the relationship between capacity (i.e., the amount a container can hold) and volume (i.e., the amount of space taken up by an object), by comparing the volume of an object with the amount of liquid it can contain or displace (e.g., a bottle has a volume, the space it takes up, and a capacity, the amount of liquid it can hold)

5.M2.08

Determine, through investigation using stacked congruent rectangular layers of concrete materials, the relationship between the height, the area of the base, and the volume of a rectangular prism, and generalize to develop the formula (i.e., Volume = area of base x height)

Volume and capacity

Volume