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 |
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 |
Hectares
So far we have looked at units of volume and area that are based of our metric units of length, millimetres, centimetres, metres and kilometres.
But there is one special unit of area that does not quite fit the pattern of the others. It is a very useful measurement that helps fill the gap between a square metre (roughly the size of a desk) and a square kilometre (the size of several city blocks).
That's where the hectare comes in.
A hectare is square $100$100 m by $100$100 m - it helps to think of it as two soccer fields side by side. This means we can convert from hectares to square metres using the following conversion.
$1$1 hectare $=$= $100$100 metres $\times$×$100$100 metres $=$=$10000$10000 square metres
Using hectares
Hectares are useful for describing area for things like sports fields, farms and parks. To be able to remember the size of a hectare, it helps to see how it is derived, and how it compares to the square metre and square kilometre.
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.
Some numbers are very large, like the distance between planets or how far light can travel in a second. Some numbers are very small like the size of an atom, or the reaction time of a robot. All of these measurements have units, and hence all also have units.
We already know how to convert units of length, area, capacity, mass and volume. Now it's time to do these conversions using very large or very small numbers.
One of the easiest ways to do this, is using scientific notation, which we have already covered in Chapter 1.
Scientific notation requires us to use a particular number of significant figures and a multiplication by a power of 10. This helps us write very large and very small numbers, using only a few digits and symbols.
Example 4
$123958372=1.24\times10^8$123958372=1.24×108 in $3$3 significant figures
$0.0000029212=2.9\times10^{-6}$0.0000029212=2.9×10−6 in $2$2 significant figures
One of the easiest ways to connect all these ideas together is through an example, let's try a question.
Example 5
$0.0000621$0.0000621 light years is said to be the distance between Jupiter and the Earth.
a) Write this in scientific notation
b) Using the value $9.4605284\times10^{12}$9.4605284×1012 kilometres in one light year, how many kilometres is it between Jupiter and Earth, to $3$3 significant figures.
c) How many metres is this?
Think: to answer this question we need to remember how to write using scientific notation, how to multiply using scientific notation, and how many metres there are in a kilometre ($1000$1000).
Do:
a) $0.0000621$0.0000621 is $6.21\times10^{-5}$6.21×10−5
b) $6.21\times10^{-5}\times9.4605284\times10^{12}=5.87\times10^8$6.21×10−5×9.4605284×1012=5.87×108 km
c) $5.87\times10^8$5.87×108 km = $5.87\times10^8\times10^3$5.87×108×103 metres (convert to metres)
= $5.87\times10^{11}$5.87×1011 m
Practice Questions
QUESTION 4
A light year is defined as the distance that light can travel in one year. It is measured to be $9460730000000000$9460730000000000 metres.
Write this using scientific notation.
How many kilometres is this? Write this using scientific notation.
How many centimetres is this? Write this using scientific notation.
QUESTION 5
A micrometre (µm) is defined as being a millionth of a metre. This means that $1$1 µm is $0.000001$0.000001 m.
The size of a fog, mist or cloud water droplet is approximately $10$10 µm. How many would fit in a $9$9 cm sample?
Round your answer to the nearest whole unit.
QUESTION 6
How many millimetres is that?
What is this when written using scientific notation?
Metric prefixes
When we want to know how long something is we often get a numerical answer, like $10$10 m, with a unit after the number showing us the type and size of the measurement. In this case it is metres so we know it is about distance, and we have an idea about how big $1$1 metre is. If we were to ask about the length of a river it might be $9000$9000 m long. In this case it might be easier to talk about it as being $9$9 km long. Similarly , if we are measuring an insect its size might be $0.002$0.002 m in length. So we might write this as $2$2 mm instead.
| $9000$9000 m | $=$= | $9\times10^3$9×103 m | $=$= | $9$9 km |
| $0.002$0.002 m | $=$= | $2\times10^{-3}$2×10−3 m | $=$= | $2$2 mm |
All of these units are related to the metre (m). We added prefixes ("k" and "m") to this unit ("m") to make the unit bigger or smaller by a certain amount. This makes it easier to talk about values that are very large or very small.
Below is the table of all the currently approved metric prefixes. Many of these are rarely used
| Prefix | Letter | Size |
|---|---|---|
| $\text{yotta}$yotta | $\text{Y}$Y | $10^{24}$1024 |
| $\text{zetta}$zetta | $\text{Z}$Z | $10^{21}$1021 |
| $\text{exa}$exa | $\text{E}$E | $10^{18}$1018 |
| $\text{peta}$peta | $\text{P}$P | $10^{15}$1015 |
| $\text{tera}$tera | $\text{T}$T | $10^{12}$1012 |
| $\text{giga}$giga | $\text{G}$G | $10^9$109 |
| $\text{mega}$mega | $\text{M}$M | $10^6$106 |
| $\text{kilo}$kilo | $\text{k}$k | $10^3$103 |
| $\text{hecto}$hecto | $\text{h}$h | $10^2$102 |
| $\text{deca}$deca | $\text{da}$da | $10^1$101 |
| $\text{ }$ | $\text{ }$ | $10^0$100 |
| $\text{deci}$deci | $\text{d}$d | $10^{-1}$10−1 |
| $\text{centi}$centi | $\text{c}$c | $10^{-2}$10−2 |
| $\text{milli}$milli | $\text{m}$m | $10^{-3}$10−3 |
| $\text{micro}$micro | $\text{μ}$μ | $10^{-6}$10−6 |
| $\text{nano}$nano | $\text{n}$n | $10^{-9}$10−9 |
| $\text{pico}$pico | $\text{p}$p | $10^{-12}$10−12 |
| $\text{femto}$femto | $\text{f}$f | $10^{-15}$10−15 |
| $\text{atto}$atto | $\text{a}$a | $10^{-18}$10−18 |
| $\text{zepto}$zepto | $\text{z}$z | $10^{-21}$10−21 |
| $\text{yocto}$yocto | $\text{y}$y | $10^{-24}$10−24 |
Each row in the table contains a prefix which is how we would write it out "femto", a letter prefix is added to the unit "f", and the size says how much smaller that unit is compared to the original unit. For a metre we would have "femtometre", "fm" with $1\text{fm}=1\times10^{-24}m$1fm=1×10−24m.
The units commonly used with these prefixes include grams (g), litres (L), metres (m), seconds (s), kelvin (K), joules (J) and watts (W). Note that some combinations of units and prefixes are rarely used. For example, hectograms (hg), megagrams (Mg) and gigametres (gm) are not widely used. Other combinations have other names commonly used such as a micrometre (µm) being referred to as a micron.
The most common prefixes used are in the middle of the table (although deca and hecto are also fairly rare). The top half of the tables shows prefixes which make the unit bigger. The bottom half shows the prefixes that make the unit smaller. We can also see that the larger prefixes (after kilo) have capital letters as their prefix.
To convert between units with a different prefix it is easiest to reduce it to the original unit.
Worked examples
Example 6
Write $9$9 nm in metres.
Think: We need a way to convert "nm" to metres. The prefix "n" in the table shows us that this is the "nano" prefix.
| Prefix | Letter | Size |
|---|---|---|
| $\text{nano}$nano | $\text{n}$n | $10^{-9}$10−9 |
Do: From the table we can rewrite $9$9 nm as $9\times10^{-9}$9×10−9 m.
Reflect: We can think of the unit containing something in the form $10^n$10n that we can separate out to get the value in the base unit.
Example 7
What is $3.4$3.4 TL written in kilolitres?
Think: The first unit used is the teralitre (TL), which has the same base unit as the kilolitre (kL). We can use the values of the units in litres to compare them.
Do: Below is the two rows from the table we are interested in:
| Prefix | Letter | Size |
|---|---|---|
| $\text{tera}$tera | $\text{T}$T | $10^{12}$1012 |
| $\text{kilo}$kilo | $\text{k}$k | $10^3$103 |
We can see that $1$1 kL is equal to $1\times10^3$1×103 L, and $3.4$3.4 TL is equal to $3.4\times10^{12}$3.4×1012 L. We can divide these two numbers to find how many kilolitres are in $3.4$3.4 teralitres.
$\frac{3.4\times10^{12}}{10^3}=3.4\times10^9$3.4×1012103=3.4×109
So $3.4$3.4 TL $=3.4\times10^9$=3.4×109 kL.
Reflect: We can also compare the sizes of the two units. The teralitre is $10^9$109 times bigger than a kilolitre. When we move from a bigger unit to a smaller unit the value will get bigger.
$3.4\times\frac{10^{12}}{10^3}=3.4\times10^9$3.4×1012103=3.4×109
Units can have prefixes added to make them larger or smaller. Below is a table of the most common prefixes:
| Prefix | Letter | Size |
|---|---|---|
| $\text{giga}$giga | $\text{G}$G | $10^9$109 |
| $\text{mega}$mega | $\text{M}$M | $10^6$106 |
| $\text{kilo}$kilo | $\text{k}$k | $10^3$103 |
| $\text{hecto}$hecto | $\text{h}$h | $10^2$102 |
| $\text{deca}$deca | $\text{da}$da | $10^1$101 |
| $\text{ }$ | $\text{ }$ | $10^0$100 |
| $\text{deci}$deci | $\text{d}$d | $10^{-1}$10−1 |
| $\text{centi}$centi | $\text{c}$c | $10^{-2}$10−2 |
| $\text{milli}$milli | $\text{m}$m | $10^{-3}$10−3 |
| $\text{micro}$micro | $\text{μ}$μ | $10^{-6}$10−6 |
| $\text{nano}$nano | $\text{n}$n | $10^{-9}$10−9 |
Practice questions
Question 7
Which of the following is the same wattage as $1000000$1000000 W?
$1$1 kW
A$1$1 µW
B$1$1 MW
C$1$1 GW
D
Question 8
Write $52$52 ms in ns.
Question 9
Write $5\times10^5$5×105 dL in hL.