Standard Units of Measure

Measure	What it measures	Common units
Length	Distance, height, perimeter	mm, cm, m, km
Area	Surface covered	mm², cm², m², km², hectares
Volume	Space occupied by a solid	mm³, cm³, m³
Capacity	How much a container holds	ml, cl, litres
Mass	How much matter (informally, weight)	mg, g, kg, tonnes
Time	Duration	seconds, minutes, hours, days
Money	Cost or value	pence, pounds

Dimensions tell you what a formula measures.
One length multiplied by another gives an area — units squared.
Three lengths multiplied give a volume — units cubed.
So a formula like $2\pi r$ (one length) must be a perimeter, while $\pi r^2$ (two lengths) must be an area.

The direction rule

Going to a smaller unit → the number gets bigger → multiply
Going to a bigger unit → the number gets smaller → divide

Measure	Conversions
Length	$1$ cm $= 10$ mm · $1$ m $= 100$ cm · $1$ km $= 1000$ m
Mass	$1$ g $= 1000$ mg · $1$ kg $= 1000$ g · $1$ tonne $= 1000$ kg
Capacity	$1$ litre $= 1000$ ml $= 100$ cl
Volume ↔ capacity	$1$ ml $= 1\text{ cm}^3$ · $1$ litre $= 1000\text{ cm}^3$ · $1\text{ m}^3 = 1000$ litres
Time	$1$ min $= 60$ s · $1$ h $= 60$ min $= 3600$ s · $1$ day $= 24$ h

Worked Example 1 — A two-step conversion

Convert $2350$ mm into metres.

①mm → cm: $2350 \div 10 = 235$ cm

②cm → m: $235 \div 100 = 2.35$ m

Do it one step at a time along the chain — trying to jump two steps at once is where errors creep in.

$1\text{ m}^2$ is NOT $100\text{ cm}^2$. A square metre is a square of side $100$ cm, so it contains $100 \times 100 = 10\,000$ small squares.

The rule

If the length factor is $n$, the area factor is $n^2$ and the volume factor is $n^3$.

Conversion	Length	Area	Volume
cm ↔ mm	$\times 10$	$\times 100$	$\times 1000$
m ↔ cm	$\times 100$	$\times 10\,000$	$\times 1\,000\,000$
km ↔ m	$\times 1000$	$\times 1\,000\,000$	$\times 10^9$

Worked Example 2 — Converting an area

A window has area $1.8\text{ m}^2$. Convert this to $\text{cm}^2$.

①Length factor m → cm is $\times 100$, so the area factor is $100^2 = 10\,000$.

②$1.8 \times 10\,000 = 18\,000\text{ cm}^2$

Worked Example 3 — Volume and capacity

A tank measures $1.2$ m by $0.8$ m by $0.5$ m. How many litres does it hold?

①Volume $= 1.2 \times 0.8 \times 0.5 = 0.48\text{ m}^3$

②$1\text{ m}^3 = 1000$ litres

③$0.48 \times 1000 = 480$ litres

Hectares. Land area is often given in hectares. $1$ hectare $= 10\,000\text{ m}^2$ — the area of a square $100$ m by $100$ m. And $1\text{ km}^2 = 100$ hectares.

Time is not metric. $2$ hours $30$ minutes is $2.5$ hours, not $2.3$ hours. To turn minutes into a decimal of an hour, divide by $60$.

Minutes	$6$	$12$	$15$	$20$	$30$	$45$	$50$
Decimal hours	$0.1$	$0.2$	$0.25$	$0.333$	$0.5$	$0.75$	$0.833$

Worked Example 4 — Timetables

A train leaves at $14{:}37$ and arrives at $17{:}12$. How long is the journey?

①From $14{:}37$ to $15{:}00$ is $23$ minutes.

②From $15{:}00$ to $17{:}00$ is $2$ hours.

③From $17{:}00$ to $17{:}12$ is $12$ minutes.

④Total $= 2$ h $+ 23 + 12 = 2$ h $35$ min.

Counting up to the next whole hour is far safer than subtracting the two times directly.

Imperial	Metric (approx.)
$1$ inch	$2.5$ cm
$1$ foot ($12$ inches)	$30$ cm
$1$ mile	$1.6$ km (so $5$ miles $= 8$ km)
$1$ pound (lb)	$450$ g (so $1$ kg $\approx 2.2$ lb)
$1$ stone ($14$ lb)	$6.35$ kg
$1$ pint	$570$ ml
$1$ gallon ($8$ pints)	$4.5$ litres

Worked Example 5 — Miles and kilometres

A journey is $72$ km. Roughly how many miles is this? Use $5$ miles $= 8$ km.

①$72 \div 8 = 9$ lots of $8$ km.

②$9 \times 5 = 45$ miles.

Sense check: a mile is longer than a kilometre, so the number of miles should be smaller ✓

Every measurement is rounded, so the true value lies within a range.

Bounds

A value rounded to the nearest $n$ lies within $\pm \dfrac{n}{2}$ of the stated value

Worked Example 6 — Finding bounds

A length is measured as $8.6$ cm, correct to $1$ decimal place. Find the lower and upper bounds.

①Rounded to the nearest $0.1$, so the half-unit is $0.05$.

②Lower bound $= 8.6 - 0.05 = 8.55$ cm

③Upper bound $= 8.6 + 0.05 = 8.65$ cm

So the true length $L$ satisfies $8.55 \leq L \lt 8.65$.

Worked Example 7 — Bounds in a calculation

A rectangle measures $12$ cm by $7$ cm, each to the nearest centimetre. Find the upper bound for its area.

①Upper bounds: $12.5$ cm and $7.5$ cm.

②Maximum area $= 12.5 \times 7.5 = 93.75\text{ cm}^2$

(The lower bound would be $11.5 \times 6.5 = 74.75\text{ cm}^2$.)

For a maximum product use both upper bounds; for a maximum quotient use the upper bound on top and the lower bound underneath.

Direction rule

Smaller unit → bigger number → multiply, and vice versa.

Length

mm $\to$ cm $\div10$; cm $\to$ m $\div100$; m $\to$ km $\div1000$.

Area

Square the length factor: m² $\to$ cm² is $\times 10\,000$.

Volume

Cube the length factor: m³ $\to$ cm³ is $\times 10^6$.

Capacity link

$1$ ml $= 1$ cm³; $1$ litre $= 1000$ cm³; $1$ m³ $= 1000$ litres.

Hectare

$1$ ha $= 10\,000$ m²; $1$ km² $= 100$ ha.

Time

Minutes $\div 60$ for decimal hours. $2$ h $30$ $= 2.5$ h.

Imperial

$5$ miles $= 8$ km; $1$ kg $= 2.2$ lb; $1$ gallon $= 4.5$ litres.

Bounds

Add and subtract half the rounding unit.

Question 1

Convert: (a) $4.2$ km to metres, (b) $780$ g to kilograms, (c) $0.65$ litres to millilitres.

▶ Show solution

(a) $4.2 \times 1000 = 4200$ m

(b) $780 \div 1000 = 0.78$ kg

(c) $0.65 \times 1000 = 650$ ml

Question 2

A rectangle measures $1.5$ m by $80$ cm. Find its area in (a) $\text{m}^2$, (b) $\text{cm}^2$.

▶ Show solution

(a) $80$ cm $= 0.8$ m; area $= 1.5 \times 0.8 = 1.2\text{ m}^2$

(b) $1.2 \times 10\,000 = 12\,000\text{ cm}^2$

Check: $150 \times 80 = 12\,000$ ✓

Question 3

Convert $0.035\text{ m}^3$ into $\text{cm}^3$ and into litres.

▶ Show solution

$0.035 \times 1\,000\,000 = 35\,000\text{ cm}^3$

$35\,000 \div 1000 = 35$ litres

Question 4

A field is $250$ m by $160$ m. Find its area in hectares.

▶ Show solution

Area $= 250 \times 160 = 40\,000\text{ m}^2$

$1$ hectare $= 10\,000\text{ m}^2$

$40\,000 \div 10\,000 = 4$ hectares

Question 5

A film starts at $19{:}45$ and lasts $2$ hours $38$ minutes. When does it finish?

▶ Show solution

$19{:}45 + 2$ hours $= 21{:}45$

$21{:}45 + 38$ min: $15$ min takes us to $22{:}00$, leaving $23$ min.

Finish time $= \mathbf{22{:}23}$

Question 6

A recipe needs $2$ pints of milk. How many millilitres is this? Use $1$ pint $= 570$ ml.

▶ Show solution

$2 \times 570 = 1140$ ml (about $1.14$ litres).

Question 7

A parcel weighs $3.4$ kg. Express this in pounds, using $1$ kg $= 2.2$ lb.

▶ Show solution

$3.4 \times 2.2 = 7.48$ lb (about $7.5$ lb).

Question 8

A mass is given as $46$ kg to the nearest kilogram. Write down the lower and upper bounds, using inequality notation.

▶ Show solution

Rounded to the nearest $1$ kg, so the half-unit is $0.5$ kg.

Lower bound $= 45.5$ kg; upper bound $= 46.5$ kg.

$45.5 \leq m \lt 46.5$

Question 9

A cuboid box measures $30$ cm by $20$ cm by $15$ cm. Small cubes of side $5$ cm are packed inside.

(a) How many cubes fit exactly? (b) What is the box's capacity in litres?

▶ Show solution

(a) Along each edge: $30 \div 5 = 6$, $20 \div 5 = 4$, $15 \div 5 = 3$.

Number of cubes $= 6 \times 4 \times 3 = 72$.

(b) Volume $= 30 \times 20 \times 15 = 9000\text{ cm}^3$.

$9000 \div 1000 = 9$ litres.

Question 10

A rectangular garden is measured as $18$ m by $11$ m, each to the nearest metre. Turf costs £$4.20$ per square metre.

(a) Find the lower and upper bounds for the area. (b) Find the greatest possible cost of turfing the garden. (c) A gardener quotes £$850$. Explain whether this is definitely enough.

▶ Show solution

(a) Bounds on the sides: $17.5 \leq L \lt 18.5$ and $10.5 \leq W \lt 11.5$.

Lower area bound $= 17.5 \times 10.5 = 183.75\text{ m}^2$

Upper area bound $= 18.5 \times 11.5 = 212.75\text{ m}^2$

(b) Greatest cost $= 212.75 \times 4.20 = £893.55$

(c) The greatest possible cost is £$893.55$, which is more than £$850$.

So £$850$ is not definitely enough. (It would cover the smallest possible area, costing $183.75 \times 4.20 = £771.75$, but the true area could be larger.)