Introduction
When we measure the length of an object, we can express this length using distance units. The most common units of length we will encounter are:
millimetres (mm): a grain of sand has a length of around 1 to 3 mm.
- centimetres (cm): the length of an ant is around 1 cm.
metres (m): the length of a desk is around 1 to 2 m.
kilometres (km): the Sydney Harbour Bridge is just over 1 km long.
We can use any of these units to measure any object, but some units are more convenient than others. For example, say we measured a football field to be 48\,737 mm wide. While this may be accurate, it isn't very helpful in giving us a visual idea of how wide the field really is. In order to convey this distance in an easier to understand unit we can apply a unit conversion.
Unit conversion
We can express the same length in different units, and to convert from one unit to another we will make use of the following relationships:
1 \text{ km} = 1000 \text{ m}
1 \text{ m} = 100 \text{ cm}
1 \text{ cm} = 10 \text{ mm}
Many rulers and tape measures show two units at once. It is common to label the cm markings with numbers, and to have mm markings in between each number to show that there are 10 mm in each centimetre.
Examples
Example 1
Convert 6.52 centimetres to millimetres. Write your answer as a decimal.
We can express the same length in different units, and to convert from one unit to another we will make use of the following relationships:
1 \text{ km} = 1000 \text{ m}
1 \text{ m} = 100 \text{ cm}
1 \text{ cm} = 10 \text{ mm}
Conversion factors
Each unit conversion has a related conversion factor. In the example above we found that we can convert a length in cm to a length in m by dividing by 100.
Similarly, as there are 1000 m in 1 km, we can convert a length in km to a length in m by multiplying by 1000. A length in mm can be converted to a length in cm by dividing by 10. The conversions factors for common units are summarised in the image below.
Another way to think about using the conversion factor is to see that the number in the measurement changes in the opposite way to the unit. So when the unit gets bigger, the number gets smaller, and when the unit gets smaller, the number gets bigger.
Examples
Example 2
Victoria is 1.14 m tall. Joanne is 156 cm tall.
Work out the height of Victoria in centimetres.
Who is taller?
The conversions factors for common units of length are summarised below and can be used to perform conversions.
Converting units can also be useful when comparing two lengths that are given in two different units.
Mutliple conversions
We now know how to convert between adjacent units of length. What about converting from, say, millimetres to metres? Although we don't yet have a direct relationship for this conversion, we can make one from what we already know.
Using the fact that 1 \text{ m }= 100 \text{ cm}, and the fact that 1 \text{ cm }=10 \text{ mm}, we can combine these together to get the following relationship:
| \displaystyle 1 \text{ m} | \displaystyle = | \displaystyle 100 \text{ cm} | |
| \displaystyle = | \displaystyle 100 \times 10\text{ mm} | Use the fact that 1 \text{ cm }= 10\text{ mm} | |
| \displaystyle = | \displaystyle 1000 \text{ mm} | Simplify the multiplication |
Remember our football field? Let's try and make that distance make sense. So we have found that 1 \text{ m }=1000 \text{ mm}, and the conversion factor between millimetres and metres is 1000.
Examples
Example 3
Convert 512\,200 cm to km. Write your answer as a decimal.
To perform multiple conversions we may need to use multiple operations as described in the image below: