Units of measure

Metric units and what they measure

Every measurement is a number and a unit. "The table is 1.2" means nothing; "the table is 1.2 m long" is a measurement. The syllabus uses these metric units.

• Length: millimetre (mm), centimetre (cm), metre (m), kilometre (km).
• Mass: gram (g), kilogram (kg).
• Capacity (how much liquid a container holds): millilitre (ml), litre (l).
• Area: ${\text{mm}}^2$, ${\text{cm}}^2$, ${\text{m}}^2$, ${\text{km}}^2$.
• Volume: ${\text{mm}}^3$, ${\text{cm}}^3$, ${\text{m}}^3$.

The prefixes always mean the same thing: kilo means a thousand, centi means a hundredth and milli means a thousandth. So a kilometre is 1000 metres, a centimetre is $\frac{1}{100}$ of a metre, a millilitre is $\frac{1}{1000}$ of a litre and a kilogram is 1000 grams.

Choosing a sensible unit. Pick the unit that gives a number of a sensible size. The thickness of a coin is about 2 mm, the length of a classroom about 8 m and the distance between two towns about 30 km. A teaspoon holds about 5 ml and a bucket about 10 litres. An apple has a mass of about 150 g and a bag of cement about 25 kg. The area of a page is measured in ${\text{cm}}^2$, the floor of a room in ${\text{m}}^2$ and a country in ${\text{km}}^2$. Having a feel for these sizes also lets you spot an answer that cannot be right, such as a door 20 m tall.

Converting units of length, mass and capacity

The conversion facts to know are:

• $1$ cm $=10$ mm
• $1$ m $=100$ cm $=1000$ mm
• $1$ km $=1000$ m
• $1$ kg $=1000$ g
• $1$ litre $=1000$ ml

Which way to go. Changing to a smaller unit gives more of them, so multiply. Changing to a larger unit gives fewer of them, so divide. For example, $3.2$ m is $3.2\times 100=320$ cm, because each metre contains 100 centimetres. And $4500$ g is $4500÷1000=4.5$ kg.

Always ask whether the answer is sensible: a length in centimetres should be a bigger number than the same length in metres. If it is smaller, you divided when you should have multiplied.

To convert between units that are not next to each other, go through the metre (or the gram, or the litre). For example, $2.5$ km $=2500$ m $=250000$ cm.

Converting units of area

Area conversions are where most marks are lost, because the conversion factor is not the same as for length.

A square of side 1 cm has an area of $1{\text{cm}}^2$. Its sides are each 10 mm long, so it can be divided into 10 rows of 10 small squares, each $1{\text{mm}}^2$. That is $10\times 10=100$ small squares, so

$$1{\text{cm}}^2={10}^2=100{\text{mm}}^2$$

In the same way, a square metre is 100 cm by 100 cm, so

$$1{\text{m}}^2={100}^2=10000{\text{cm}}^2$$

and a square kilometre is 1000 m by 1000 m, so $1{\text{km}}^2={1000}^2=1000000{\text{m}}^2$.

The rule: for area, square the length conversion factor. Then multiply when going to a smaller unit and divide when going to a larger one, exactly as before. For example, $2.5{\text{m}}^2=2.5\times 10000=25000{\text{cm}}^2$, and $850{\text{mm}}^2=850÷100=8.5{\text{cm}}^2$.

A safe alternative, when a shape is given in one unit and the answer is wanted in another, is to convert the lengths first and then work out the area. A rectangle 150 cm by 80 cm is 1.5 m by 0.8 m, so its area is $1.5\times 0.8=1.2{\text{m}}^2$.

A $1$ cm by $1$ cm square holds $10\times 10=100$ squares of side 1 mm, so $1{\text{cm}}^2=100{\text{mm}}^2$, not 10.

Converting units of volume

Volume works the same way, one dimension further. A cube of side 1 cm is 10 mm long, 10 mm wide and 10 mm high, so it holds $10\times 10\times 10=1000$ cubes of side 1 mm:

$$1{\text{cm}}^3={10}^3=1000{\text{mm}}^3$$

A cube of side 1 m is 100 cm in each direction, so

$$1{\text{m}}^3={100}^3=1000000{\text{cm}}^3$$

The rule: for volume, cube the length conversion factor. So $0.4{\text{m}}^3=0.4\times 1000000=400000{\text{cm}}^3$, and $7200{\text{mm}}^3=7200÷1000=7.2{\text{cm}}^3$.

Summary of the factors between neighbouring units:

• length: mm to cm $\times 10$, cm to m $\times 100$, m to km $\times 1000$ (dividing when going up);
• area: $\times 100$, $\times 10000$, $\times 1000000$;
• volume: $\times 1000$ (mm to cm), $\times 1000000$ (cm to m).

Volume and capacity

Capacity and volume measure the same thing, the amount of space, so they are linked exactly:

• $1{\text{cm}}^3=1$ ml
• $1000{\text{cm}}^3=1$ litre (a litre is the volume of a cube of side 10 cm)
• $1{\text{m}}^3=1000$ litres

Why $1{\text{m}}^3$ is 1000 litres. $1{\text{m}}^3=1000000{\text{cm}}^3$, and every $1000{\text{cm}}^3$ is one litre, so $1000000÷1000=1000$ litres.

A typical question gives the dimensions of a tank, asks for its volume and then asks how many litres it holds. Work out the volume in ${\text{cm}}^3$ and divide by 1000, or work in metres and multiply the ${\text{m}}^3$ by 1000. Both routes give the same answer; choose the one with easier numbers.

Solving practical problems with units

In a practical problem the measurements are often given in different units: a tank 1.2 m long and 50 cm wide, or a recipe that uses 1.5 kg of flour and 250 g of sugar. The method is always the same.

1. Change everything into the same unit before you calculate. Multiplying metres by centimetres gives nonsense.
2. Do the calculation.
3. Convert the answer into the unit the question asks for, using the squared factor for an area and the cubed factor for a volume.
4. Interpret the answer. If the question asks how many whole glasses can be filled, round down; if it asks how many tins must be bought, round up, because you cannot buy part of a tin.

Choosing the working unit well saves effort. For a fish tank measured in centimetres, work in ${\text{cm}}^3$ and divide by 1000 at the end to get litres. For rainfall on a roof, a depth of 15 mm is $0.015$ m, so the volume comes out in ${\text{m}}^3$, and ${\text{m}}^3$ times 1000 gives litres.