8.08 Volume and capacity
Introduction
Conversions from volume to capacity is the mathematical equivalent of asking "How much water can we pour into this container?"
Capacity
Capacity is a measure of how much a container can hold. We normally associate capacity with liquids, but it can also be used as a measure of gasses and solids.
When measuring the capacity of containers for everyday use, we will often use either millilitres \text{(mL)} or litres \text{(L)}. For larger containers like bathtubs or swimming pools, we can use kilolitres \text{(kL)}.
We can convert between these units using the conversion equations:
1 \text{ L}= 1000 \text{ mL}
1 \text{ kL}= 1000 \text{ L}
Examples
Example 1
Convert 99\,000\text{ L} to \text{ kL}.
Capacity is a measure of how much liquid, gas or solid a container can hold.
Some of the conversions for capacity are:
1 \text{ L}= 1000 \text{ mL}
1 \text{ kL}= 1000 \text{ L}
Volume to capacity
When converting from volume to capacity, there is only one conversion equation that we need to use:
1 \text{ cm}^3 = 1 \text{ mL}
This tells us that every one cubic centimetre has a capacity of one millilitre. It also tells us that any conversion equations involving millilitres will work the same way for cubic centimetres.
This means we get this conversion equation for free:
1000 \text{ cm}^3 = 1 \text{ L}
Now we have a way to convert from volume to capacity, we can start finding the capacity of containers.
Examples
Example 2
A cylinder has a diameter of 12 \text{ cm} and height of 70 \text{ cm}.
Find the volume of the cylinder in cubic centimetres. Round your answer to one decimal place.
What is the capacity of the cylinder in litres? Round your answer to four decimal places.
Some of the volume to capacity conversions are:
1 \text{ cm}^3 = 1 \text{ mL}
1000 \text{ cm}^3 = 1 \text{ L}
Capacity of a cubic metre
As mentioned before, the only conversion we need in order to convert between volume and capacity is the equality 1 \text{ cm}^3 = 1 \text { mL}.
With this in mind, how many litres are there in one cubic metre?
To find how many litres there are in one cubic metres, there are two conversions we need to make. First, we want to convert our cubic metre into cubic centimetres. Once we have done this, we can then convert to litres, like so:
| \displaystyle 1\text{ m}^3 | \displaystyle = | \displaystyle 1 \text{ m}\times 1 \text{ m} \times 1 \text{ m} | Equate 1 \text { m}^3 to a cube of side length 1 \text{ m} |
| \displaystyle = | \displaystyle 100 \text{ cm}\times 100 \text{ cm} \times 100 \text{ cm} | Convert each metre to centimetres | |
| \displaystyle = | \displaystyle 100 \times 100 \times 100 \text{ cm}^3 | Combine the units | |
| \displaystyle = | \displaystyle 1\,000\,000 \text{ cm}^3 | Evaluate |
These calculations tell us that one cubic metre is equal to 1\,000\,000 cubic centimetres. Now that our volume is in cubic centimetres, we can convert to capacity. We can convert using the equation 1 \text{ L} = 1000 \text{ cm}^3.
| \displaystyle \text{Capacity} | \displaystyle = | \displaystyle \dfrac{1\,000\,000}{1000} | Divide by the conversion factor |
| \displaystyle = | \displaystyle 1000 \text{ L} | Evaluate | |
| \displaystyle = | \displaystyle 1\text{ kL} | Convert to kilolitres |
So the capacity of one cubic metre is 1000\text{ L}=1\text{ kL}.
To convert from capacity back into volume, we can simply reverse these steps.
Examples
Example 3
Kathleen is constructing a swimming pool designed to hold 34.4 kilolitres of water.
She has already decided on a base area of 8 square metres.
What will the volume of Kathleen's pool be in cubic metres?
If the depth of the pool is the same at every point, how deep must it be in metres?
1\text{ m}^3=1000\text{ L}=1\text{ kL}