10.09 Units of volume and capacity

Volume vs 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 but we can also use units such as millilitres and litres to describe volumes. Capacity uses the units for volume and given it often deals with liquids common units are millilitres and litres.

 

Units of volume

Common units for volume from smallest to largest are:

• millimetres cubed (mm³)
• centimetres cubed (cm³)
• metres cubed (m³)

 

Units of capacity

Common units for capacity from smallest to largest are:

• millilitres (mL)
• litres (L)
• kilolitres (kL)
• megalitres (ML)

The symbol for litres can use a lower case or upper case letter L, but usually the upper case L is used to avoid confusion with the number $1$1.

 

Estimating volume and capacity

Just as with lengths and areas to determine the correct unit to use, and be able to estimate the volume or capacity of an object it's useful to identify some common items that are about the size of the units above. Such as:

• $1$1 mm³ is very small! This is a cube with length, width and height $1$1 mm - about the size of $30$30 grains of salt.
• $1$1 cm³ is the size of a cube with length, width and height $1$1 cm - this is about two m&ms. A teaspoon holds $5$5 cm³.
• $1$1 m³ is the size of a cube with length, width and height $1$1 m - about the size of a fridge.
• $1$1 mL is the same as $1$1 cm³ and a teaspoon is $5$5 mL.
• $1$1 L. It is very common for milk to come in $1$1 and $2$2 litre cartons.
• $1$1 kL is one thousand litres. This is equal to $1$1 m³, and is around $6$6 bathtubs full of water.
• $1$1 ML is one million litres - this is a lot of liquid! An Olympic swimming pool holds on average $2.5$2.5 ML.

Sometimes we need to estimate the volume or capacity of an object and when estimating it is useful to have some points of reference. Consider the items above together with some other common items, such as those in the video below, as points for comparison in making estimates.

The following video shows some common items and units of capacity.

 

Practice questions

Question 1

Question 2

Question 3

 

Converting units of volume

The following diagram shows the conversion factor between different units of volume.

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".

However, 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 m³ to cm³ is $1000000$1000000, which can be remembered as multiplying by conversion from metres to centimetres three times.

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

 

Converting units of capacity

The following diagram shows the conversion factor between common units for capacity:

Note: The conversion factor is $1000$1000 at each step.

 

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 is this 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

Question 5

Question 6