When measuring the length, area, or volume of an object, we want to choose the most appropriate unit. Sometimes the units we are given are not the easiest to deal with, they may be too large or too small and make calculations cumbersome.
A blue whale is $31456$31456 mm long, a one dollar coin has an area of $0.000491$0.000491 square metres, and an apple has a volume of $0.00024$0.00024 cubic metres. These are all correct measurements, but are they useful? Can you visualise these sizes in your mind?
Now, what if you were told a blue whale is about $32$32 m long, a one dollar coin has an area of $491$491 mm2, which is a little bit less than $5$5 square centimetres, or that an apple has a volume of $240$240 cubic centimetres. These numbers are much more sensible, can make it much easier to visualise and also to make comparisons to other sizes. Sometimes, as in the case of the coin, two different units might both be considered appropriate.
Unit conversion
Much like we can express the same length in different units, we can convert from one unit of area to another, and similarly from one unit of volume to another we will make use of the following relationships. Let's first remind ourselves of the length conversions, as we will use these to work out the conversions for area and volume.
| $1$1 | km | $=$= | $1000$1000 | m | $=$= | $100000$100000 cm |
| $1$1 | m | $=$= | $100$100 | cm | $=$= | $10000$10000 mm |
| $1$1 | cm | $=$= | $10$10 | mm | $=$= | $0.01$0.01 m |
| Metres | Visual representation | Centimetres | |
|---|---|---|---|
| Length (one dimension) | $1\text{ metre}$1 metre |  |
$1\text{ m}=100\text{ cm}$1 m=100 cm |
| Area (two dimensions) | $1\text{ square metre}=1\text{ m}\times1\text{ m}$1 square metre=1 m×1 m |   |
$1\text{ m}^2=100\text{ cm}\times100\text{ cm}=10000\text{ cm}^2$1 m2=100 cm×100 cm=10000 cm2 |
| Volume (three dimensions) | $1\text{ cubic metre}=1\text{ m}\times1\text{ m}\times1\text{ m}$1 cubic metre=1 m×1 m×1 m |  |
$1\text{ m}^3=100\text{ cm}\times100\text{ cm}\times100\text{ cm}=1000000\text{ cm}^3$1 m3=100 cm×100 cm×100 cm=1000000 cm3 |
Worked Examples
Example 1
Convert $0.0023$0.0023 kilometres into centimetres.
Think: We are converting a distance in kilometres into a distance in centimetres, so we want to use the relationship $1$1 km $=$=$100000$100000 cm.
Do: We can multiply both sides of the equality by $0.0023$0.0023 to get:
| $0.0023\times1$0.0023×1 km | $=$= | $0.023\times100000$0.023×100000 m |
| $0.0023$0.0023 km | $=$= | $230$230 cm |
So $0.0023$0.0023 kilometres can be converted into $230$230 centimetres.
Example 2
Convert $3700$3700 square millimetres into square centimetres.
Think: We are converting square millimetres into square centimetres . We know that for length the relationship is $1$1 cm $=$=$10$10 mm, but what about for square units? Well $1$1 square centimetre is a square with a side length of $1$1 cm, and we know that equals $10$10 mm.
This means $1$1 cm $\times$× $1$1 cm $=$= $10$10 mm $\times$× $10$10 mm which gives us $100$100 mm2.
Giving us the relationship $1$1 cm2 = $100$100 mm2.
If $1$1 cm$=$=$100$100 mm2 then $1$1 mm2 $=$= $\frac{1}{100}$1100 cm2. We can divide $3700$3700 by $100$100 to find how many square centimetres it is.
Do:
| $3700$3700 mm2 | $=$= | $\frac{3700}{100}$3700100 cm2 |
| $3700$3700 mm2 | $=$= | $37$37 cm2 |
So $3700$3700 mm2 is equal to $37$37 cm2.
Reflect: By reversing the conversion relationship, we can divide by the conversion factor when we want to convert from a smaller unit to a larger one.
In these two examples we found that we could convert between area units by multiplying by the conversion factor when we want to go from the larger unit to the smaller unit, and divide by conversion factor when going from a smaller unit to the larger unit. We also found that we could convert between area units by multiplying by the length factor two times, as we were measuring a square instead of a line.
We can extend this idea further to find the conversions for volume. Volume measures how many cubes of a particular size fill a space, and so we can convert between units of volume by multiplying by the length factor three times!
Example 3
Convert $420000$420000 cubic centimetres into cubic metres.
Think: Since there are $100$100 cm in $1$1 m, there must be $100\times100\times100$100×100×100 cubic centimetres in one cubic metre. This is because a cubic metre is a cube with dimensions of $100$100 cm $\times100$×100 cm $\times100$×100 cm = $1000000$1000000 cm3. So let's use this relationship. We are converting from a smaller unit to a larger unit so we will divide by the conversion factor of $1000000$1000000.
Do:
| $420000$420000 cm3 | $=$= | $42000\div1000000$42000÷1000000 cm3 |
| $420000$420000 cm3 | $=$= | $0.42$0.42 m3 |
So $420000$420000 cubic centimetres can be converted into $0.42$0.42 cubic metres.
Reflect: The two numbers on the final line are different only by a factor of $1000000$1000000. This factor comes from applying the original relationship $100$100 cm $=$=$1$1 m three times, once for each dimension. Once we understand the process outlined above, we can use these conversion factors to convert between any units.
Area and volume conversions
| $1$1 | km2 | $=$= | $1000000$1000000 | m2 |
| $1$1 | m2 | $=$= | $10000$10000 | cm2 |
| $1$1 | cm2 | $=$= | $100$100 | mm2 |
| $1$1 | m3 | $=$= | $1000000$1000000 | cm3 |
| $1$1 | cm3 | $=$= | $1000$1000 | mm3 |
Practice Questions
Question 1
Select the option that shows $6$6 km2 converted into m2.
$60000$60000 m2
A$0.000006$0.000006 m2
B$6000$6000 m2
C$6000000$6000000 m2
D
Question 2
The rectangle below has side lengths given in cm.
Convert the dimensions of the rectangle into metres:
$500$500 cm $=$= $\editable{}$ m $385$385 cm $=$= $\editable{}$ m Hence find the area of the rectangle in square metres.
Question 3
Convert $9.77$9.77 cm3 to mm3.