8.06 Units of volume and capacity
Volume is the amount of space an objects takes up. This can be the amount of space a 3D shape occupies, or the space that a substance (solid, liquid or gas) fills. Capacity, on the other hand, is the amount a container can hold, rather than the amount of space the container itself displaces. Capacity will only be used in relation to a container and often involves liquids.
The units for volume are generally cubic units such as mm3, cm3 and m3. Capacity often deals with liquids, and so the common units are millilitres (mL), litres (L) and kilolitres (kL).
Units of volume
We use the notion of cubic units to measure volume. A cubic unit is the volume of a cube with a side length of one unit.
The units for length include millimetres, centimetres, metres and kilometres so we end up with the following units for volume.
| Volume units | Symbol | Description |
|---|---|---|
|
cubic millimetres |
mm3 |
The volume of a cube with side lengths of $1$1 mm each. About the size of a grain of sand. |
|
cubic centimetres |
cm3 |
A cube with side lengths of $1$1 cm each. The size of a sugar cube. |
|
cubic metres |
m3 |
A cube with side lengths of $1$1 m each. About the size of a loaded shipping pallet. |
|
cubic kilometres |
km3 |
A cube with a side length of $1$1 km. This is a very large volume, used to describe the amount of water in an ocean or very large lake. |
When converting units of volume, we need to work out how many smaller cubic units fit into the larger cubic unit.
The conversion factors for volume involve very large numbers, so index notation is used in this conversion chart. $10^3=1000$103=1000, or "$1$1 followed by $3$3 zeros".
To convert to smaller units multiply by the conversion factor.
To convert to bigger units divide by the conversion factor.
Note: Rather than remembering these factors it is useful to understand that to convert units of volume we are converting the three length dimensions of a cube unit. We can do this by multiplying or dividing by the conversion factor for lengths three times - that is multiply or divide by the conversion factor cubed. This means we only have to recall the conversion factor for lengths.
Worked example
Example 1
Convert $5.85$5.85 m3 into cm3.
Think: Think about the steps needed to move from m3 to cm3. We are going from large units to small units, so we need to multiply. The conversion factor from m to cm is $100$100, therefore the conversion factor for m3 to cm3 is $100$1003 or $1000000$1000000
Do:
| $5.85$5.85 m3 | $=$= | $5.85\times100\times100\times100$5.85×100×100×100 cm3 |
| $=$= | $5.85\times1000000$5.85×1000000 cm3 | |
| $=$= | $5850000$5850000 cm3 |
Practice questions
Question 1
Convert $3.5$3.5 cm3 to mm3.
Question 2
Complete the working below to convert $460000000$460000000 mm3 to a volume in m3.
$460000000$460000000 mm3 $=$= $460000000$460000000$\div$÷$1000$1000 cm3 (there are $1000$1000 mm3 in each cm3) $=$= $\editable{}$ cm3 $=$= $\editable{}$$\div$÷$1000000$1000000 m3 (there are $1000000$1000000 cm3 in each m3) $=$= $\editable{}$ m3
Units of capacity
We use capacity when we are concerned with the quantity of a fluid (i.e. a liquid or gas) that a container can hold.
The common units for capacity from smallest to largest are:
- millilitres (mL)
- litres (L)
- kilolitres (kL)
- megalitres (ML)
We use the upper case letter "L" for litres to avoid confusion with the number $1$1.
| Capacity units | Symbol | Description |
|---|---|---|
|
millilitres |
mL |
Equivalent to $1$1 cm3 |
|
litres |
L |
Equivalent to $1000$1000 cm3 |
|
kilolitres |
kL |
Equivalent to $1$1 m3 |
|
megalitres kilometres |
ML |
Equivalent to $1000$1000 m3 |
When converting units of capacity, we need to work out how many smaller units fit into the larger unit.
To convert to smaller units multiply by the conversion factor.
To convert to bigger units divide by the conversion factor.
Note: The conversion factor is $1000$1000 at each step.
Practice question
Question 3
Convert $1.54$1.54 litres to millilitres.
$1.54$1.54 litres = $\editable{}$ millilitres
Converting between units
In some cases, we may want to convert from units of volume to units of capacity, or vice-versa. For example, to find the capacity of a swimming pool, it would be easier to first measure the dimensions of the pool in metres and then convert to litres. In the diagram below we can see that $1$1 cm3 is equivalent to $1$1 mL and that $1$1 m3 is equivalent to $1000$1000 L.
Worked example
example 2
A fish tank is $350000$350000 cm3. What volume of water in litres is required to fill the fish tank?
Think: Think about the steps needed to move from cm3 to litres. (cm3 $\rightarrow$→ mL $\rightarrow$→ L). We know that $1$1 cm3 is equivalent to $1$1 mL, so first write the amount in mL and then convert to litres using the conversion factor or $1000$1000.
Do: First convert to millilitres: $350000$350000 cm3$=$=$350000$350000 mL
Now convert to litres: $350000$350000 mL$\div$÷$1000=350$1000=350 L
Practice questions
Question 4
Convert $40$40 millilitres (mL) to cubic centimetres (cm3 ).
Question 5
Convert $6750$6750 cubic centimetres (cm3) to litres (L).
Question 6
Consider a volume of $72000$72000 cm3.
Convert this volume to a capacity in L.
Now convert this to a capacity in kL.