Standard Units

• length: millimetres (mm), centimetres (cm), metres (m), kilometres (km);
• area: square millimetres (mm²), square centimetres (cm²), square metres (m²), square kilometres (km²) and hectares (ha);
• volume: cubic millimetres (mm³), cubic centimetres (cm³) and cubic metres (m³);
• capacity: millilitres (ml), centilitres (cl) and litres (l);
• mass: milligrams (mg), grams (g), kilograms (kg) and tonnes (t);
• time: seconds (s), minutes (min), hours (h) and days;
• money: pence (p) and pounds (£).

Choose a sensible-sized unit. Millimetres suit the thickness of a coin, metres suit a room and kilometres suit a long journey. The object has not changed when its unit changes: $2.35\text{ m}$ and $2350\text{ mm}$ represent the same length.

Converting linear metric units

The main length relationships are

Moving to a smaller unit makes the numerical value larger, so multiply. Moving to a larger unit makes the numerical value smaller, so divide. For example,

and

A conversion can cross more than one step. Since $1\text{ m}=1000\text{ mm}$, converting $2350\text{ mm}$ to metres gives $2350\div1000=2.35\text{ m}$. Writing the conversion relationship before calculating is safer than relying on a remembered instruction such as “move the decimal point”.

The corresponding mass relationships include

Thus $0.72\text{ kg}=720\text{ g}$, whereas $4500\text{ g}=4.5\text{ kg}$.

Why area conversions use squared factors

Area measures surface covered, so its units are squared. A square with side $1\text{ m}$ also has side $100\text{ cm}$. Its area is therefore

Consequently,

not $100\text{ cm}^2$. In general, if the length conversion factor is $n$, the area conversion factor is $n^2$. For instance, metres to centimetres has length factor $100$, so square metres to square centimetres has factor $100^2=10\,000$.

Therefore $1.8\text{ m}^2=1.8\times10\,000=18\,000\text{ cm}^2$. In the reverse direction, $35\,000\text{ cm}^2\div10\,000=3.5\text{ m}^2$.

For land area,

which is the area of a $100\text{ m}$ by $100\text{ m}$ square. Also, $1\text{ km}^2=100$ hectares.

Why volume conversions use cubed factors

Volume measures three-dimensional space, so its units are cubed. A cube with edge $1\text{ m}$ has edges of $100\text{ cm}$, giving

In general, if the length factor is $n$, the volume factor is $n^3$. Hence $0.035\text{ m}^3=0.035\times1\,000\,000=35\,000\text{ cm}^3$.

The units themselves show why the factor is cubed:

Connecting volume and capacity

Volume and capacity describe closely related quantities: volume is the space occupied, while capacity is how much a container can hold. Important links are

A tank with volume $0.48\text{ m}^3$ therefore holds $0.48\times1000=480$ litres. Similarly, $9000\text{ cm}^3=9000\text{ ml}=9$ litres.

Time does not follow metric place value

Time conversions use factors of $60$ and $24$:

To express minutes as part of an hour, divide by $60$. Thus $30$ minutes is $30\div60=0.5$ hours, so $2$ hours $30$ minutes is $2.5$ hours. It is not $2.3$ hours because $30$ minutes is not three-tenths of an hour.

For elapsed-time questions, counting forwards can avoid difficulties at the change of hour. From 14:37 to 15:00 is 23 minutes, from 15:00 to 17:00 is 2 hours, and from 17:00 to 17:12 is 12 minutes. The total is therefore 2 hours 35 minutes.

Converting money

UK money uses

Pounds to pence requires multiplication by $100$, so £$4.35=435$p. Pence to pounds requires division by $100$, so $780$p = £$7.80$. Money should normally be written to two decimal places when expressed in pounds: £$6.5$ is conventionally written as £$6.50$.

Units in algebraic contexts

The same conversion factors apply when a measurement contains a variable. The letter represents a number; it does not prevent the conversion. If a length is $x$ metres, then

If a mass is $m$ kilograms, then

Area and volume still require squared and cubed factors:

You may also need to simplify an expression after converting. For example, a ribbon of length $(3x+2)$ metres has length

Convert the whole expression, not just one term.

Units must be compatible before values are added, subtracted or substituted into a formula. If a rectangle has length $a$ metres and width $40$ centimetres, first rewrite $40\text{ cm}$ as $0.4\text{ m}$. Its area is then

Units also help identify what a formula represents. Multiplying two lengths produces square units and therefore an area; multiplying three lengths produces cubic units and therefore a volume. Keeping the units visible is a useful check on both the calculation and the final answer.