10.01 Units of length

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$1 to $3$3 mm.
• centimetres (cm): the length of an ant is around $1$1 cm.
• metres (m): the length of a desk is around $1$1 to $2$2 m.
• kilometres (km): the Sydney Harbour Bridge is just over $1$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 $48737$48737 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$1	km	$=$=	$1000$1000	m
$1$1	m	$=$=	$100$100	cm
$1$1	cm	$=$=	$10$10	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$10 mm in each centimetre.

Worked examples

Example 1

Convert $3$3 kilometres into metres.

Think: We are converting a distance in kilometres into a distance in metres, so we want to use the relationship $1$1 km $=$=$1000$1000 m.

Do: We can multiply both sides of the equality by $3$3 to get:

$3\times1$3×1 km	$=$=	$3\times1000$3×1000 m
$3$3 km	$=$=	$3000$3000 m

So $3$3 kilometres can be converted into $3000$3000 metres.

example 2

Convert $260$260 millimetres into centimetres.

Think: We are converting millimetres into centimetres so we want to use the relationship $1$1 cm $=$=$10$10 mm.

Do: We can multiply both sides of the equality by $26$26 to get:

$26\times1$26×1 cm	$=$=	$26\times10$26×10 mm
$26$26 cm	$=$=	$260$260 mm

So $260$260 millimetres can be converted into $26$26 centimetres.

In each of these examples we treated the equality relationship like a ratio and multiplied both sides by the same amount. We can also convert units by dividing both sides of the equality.

example 3

Convert $43$43 centimetres into metres.

Think: Since there are $100$100 cm in $1$1 m, we know that $43$43 cm will be less than $1$1 m. If we can find a way to write $1$1 cm in terms of metres, then we can use this to write $43$43 cm in terms of m.

Do: Begin with the relationship between cm and m:

$100$100 cm	$=$=	$1$1 m	(Start with the relationship)
$\frac{100}{100}$100100​ cm	$=$=	$\frac{1}{100}$1100​ m	(Divide both sides by $100$100)
$1$1 cm	$=$=	$0.01$0.01 m	(Simplify both sides)
$43\times1$43×1 cm	$=$=	$43\times0.01$43×0.01 m	(Multiply both sides by $43$43)
$43$43 cm	$=$=	$0.43$0.43 m	(Simplify both sides)

So $43$43 centimetres can be converted into $0.43$0.43 metres.

Reflect: The two numbers on the final line are different only by a factor of $100$100. That is, $\frac{43}{100}=0.43$43100​=0.43, and $43=0.43\times100$43=0.43×100. This factor comes from the relationship $100$100 cm $=$=$1$1 m. Once we understand the process outlined above, we can use this factor to quickly convert between centimetres and metres.

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$100.

Similarly, as there are $1000$1000 m in $1$1 km, we can convert a length in km to a length in m by multiplying by $1000$1000. A length in mm can be converted to a length in cm by dividing by $10$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.

Worked example

example 4

Convert $3.47$3.47 kilometres into metres.

Think: What relationship should we use? What is the conversion factor? Are the units getting bigger or smaller?

Do: Since we are converting between kilometres and metres we want to use the relationship $1$1 km $=$=$1000$1000 m, which has a conversion factor of $1000$1000.

When converting kilometres to metres, the unit gets smaller ($1$1 m is a smaller length than $1$1 km). To balance this out the number must get bigger, so this means we need to multiply by the conversion factor. Putting all of this together we have:

$3.47$3.47 km	$=$=	$3.47\times1000$3.47×1000 m
	$=$=	$3470$3470 m

So $3.47$3.47 kilometres can be converted into $3470$3470 metres.

Multiple 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$1 m $=$=$100$100 cm, and the fact that $1$1 cm $=$=$10$10 mm, we can combine these together to get the following relationship:

$1$1 m	$=$=	$100$100 cm
	$=$=	$100\times10$100×10 mm	(Use the fact that $1$1 cm $=$=$10$10 mm)
	$=$=	$1000$1000 mm	(Simplify the multiplication)

So we have found that $1$1 m $=$=$1000$1000 mm, and the conversion factor between millimetres and metres is $1000$1000.

Remember our football field? Let's try and make that distance make sense!

Worked example

example 5

A football field has a width of $48737$48737 millimetres. How wide is the football field in metres?

Think: To convert from millimetres to metres we want to use the relationship $1$1 m $=$=$1000$1000 mm. Since metres are bigger than millimetres, we want to divide the number by the conversion factor.

Do: To convert the width we will divide by $1000$1000. This gives us:

Width of the football field	$=$=	$48737$48737 mm
	$=$=	$\frac{48737}{1000}$487371000​ m
	$=$=	$48.737$48.737 m

We can now see that a football field is approximately $50$50 metres long. Now that's a number that we can understand!

Reflect: We found the required relationship by combining two unit conversions into one. We then applied it to the dimension of the field by changing the units and the measurement accordingly.

This last example shows that converting units is something we can do when we want to make a number easier to understand. It can also be useful when comparing two lengths that are given in two different units.

Practice questions

Question 1

Question 2

Question 3