Units of Measurements

The metric system is built on powers of 10, which makes converting straightforward. The facts you build on this year:

• Length: $10\text{ mm} = 1\text{ cm}$, $100\text{ cm} = 1\text{ m}$, $1000\text{ m} = 1\text{ km}$.
• Mass: $1000\text{ g} = 1\text{ kg}$, $1000\text{ kg} = 1\text{ tonne}$.
• Capacity: $1000\text{ ml} = 1\text{ litre}$.

So $2.4\text{ km} = 2.4 \times 1000 = 2400\text{ m}$, and $850\text{ g} = 850 \div 1000 = 0.85\text{ kg}$.

This year the conversions get trickier in two ways: area and volume units use different conversion numbers, and you handle the accuracy of measurements much more carefully. Both build on this same multiply-or-divide thinking.

There are $100\text{ cm}$ in a metre — but there are not $100\text{ cm}^2$ in a square metre. A square of side $1\text{ m}$ is $100\text{ cm}$ along each side, so its area is $100 \times 100 = 10\,000\text{ cm}^2$.

So the area rule is to square the length conversion factor:

• $1\text{ cm}^2 = 10 \times 10 = 100\text{ mm}^2$
• $1\text{ m}^2 = 100 \times 100 = 10\,000\text{ cm}^2$

Volume units go a step further — you cube the factor, because volume has three dimensions:

• $1\text{ cm}^3 = 10 \times 10 \times 10 = 1000\text{ mm}^3$
• $1\text{ m}^3 = 100 \times 100 \times 100 = 1\,000\,000\text{ cm}^3$

A handy capacity link: $1\text{ cm}^3$ of space holds exactly $1\text{ ml}$, so $1000\text{ cm}^3 = 1\text{ litre}$. When a volume problem mixes units, convert everything to the same unit first.

A compound unit is built from two measurements — it describes a rate, comparing one quantity with another. The word "per" in the unit always means "divided by".

You meet three this year:

• Speed combines distance and time: $\text{speed} = \dfrac{\text{distance}}{\text{time}}$, in units like $\text{km/h}$ or $\text{m/s}$.
• Density combines mass and volume: $\text{density} = \dfrac{\text{mass}}{\text{volume}}$, in units like $\text{g/cm}^3$.
• Pressure combines force and area: $\text{pressure} = \dfrac{\text{force}}{\text{area}}$, in units like $\text{N/cm}^2$.

The triangle trick works for all three. Cover the quantity you want and read off the rest. For pressure: cover F and you see $\text{P} \times \text{A}$; cover A and you see $\text{F} \div \text{P}$.

So a force of $48\text{ N}$ spread over an area of $6\text{ cm}^2$ gives a pressure of $48 \div 6 = 8\text{ N/cm}^2$. A block of mass $90\text{ g}$ and volume $30\text{ cm}^3$ has density $90 \div 30 = 3\text{ g/cm}^3$.

Many problems do not ask for the rate itself — they give you the rate and ask for one of the other quantities. The reliable method:

1. Write down what you have and what you want.
2. Use the triangle to choose the right form of the relationship.
3. Substitute the numbers and work it out, keeping units consistent.

Suppose a cyclist travels at $24\text{ km/h}$ for $45$ minutes, and you want the distance. Cover D in the triangle and you see $\text{S} \times \text{T}$. But first the time must be in hours: $45\text{ minutes} = 45 \div 60 = 0.75\text{ hours}$. Then: $\text{distance} = 24 \times 0.75 = 18\text{ km}.$

Density and pressure rearrange the same way. To find a mass from a density, use $\text{mass} = \text{density} \times \text{volume}$. To find an area from a pressure, use $\text{area} = \text{force} \div \text{pressure}$. The triangle is the same idea every time.

No real measurement is perfectly exact, so a measurement is always given to a degree of accuracy — for example "to the nearest centimetre". That means the true value lies somewhere in a small range.

The two ends of the range are the lower bound and the upper bound:

• The lower bound is half a unit below the rounded value.
• The upper bound is half a unit above the rounded value.

A length given as $8\text{ cm}$ to the nearest centimetre has a lower bound of $7.5\text{ cm}$ and an upper bound of $8.5\text{ cm}$. The true length is anywhere from $7.5\text{ cm}$ up to (but not quite reaching) $8.5\text{ cm}$.

The "half a unit" depends on the rounding. To the nearest $10$, a mass of $40\text{ kg}$ has bounds of $35\text{ kg}$ and $45\text{ kg}$. To one decimal place, a time of $3.2\text{ s}$ has bounds of $3.15\text{ s}$ and $3.25\text{ s}$.

Getting confident with units makes a lot of later work smoother:

• Area and volume rely on converting square and cube units correctly.
• Speed, density and pressure problems appear in both maths and science.
• Accuracy and bounds lead into error in measurement and sensible rounding of answers.
• In everyday life, units run through cooking, travel, sport, engineering and science experiments.

If a measurement question feels tricky later, it is often a unit conversion or an accuracy decision underneath. Come back to this guide whenever you need it — careful unit work pays off everywhere.