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C5.1 Units of Measure

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01 — Explanation

Why do area and volume conversions use squared and cubed factors?

The metric system scales every unit by powers of $10$ , so converting between units always means multiplying or dividing by $10$ , $100$ or $1000$ . But the power you use depends on whether you're converting a length (1D), an area (2D), or a volume (3D) — an area is made of two lengths multiplied together, so its conversion factor gets squared; a volume is three lengths multiplied together, so its factor gets cubed. Getting this wrong is one of the most common ways marks are lost in mensuration questions, so it's worth pausing on every conversion to ask: is this a length, an area, or a volume?

02 — Explanation

How do I convert between length units?

10\text{ mm} = 1\text{ cm} \qquad 100\text{ cm} = 1\text{ m} \qquad 1000\text{ m} = 1\text{ km}

Worked example. Convert $4.5\text{ km}$ into millimetres.

Length conversions are done one step at a time to avoid arithmetic slips:

4.5\text{ km} = 4.5\times1000 = 4500\text{ m} \qquad 4500\text{ m} = 4500\times100 = 450\,000\text{ cm} \qquad 450\,000\text{ cm}=450\,000\times10=4\,500\,000\text{ mm}

03 — Explanation

How do I convert between area units?

Square the length conversion factor:

1\text{ m}^2 = 100^2\text{ cm}^2 = 10\,000\text{ cm}^2 \qquad 1\text{ cm}^2 = 10^2\text{ mm}^2 = 100\text{ mm}^2

Worked example. A tile has area $3.2\text{ m}^2$ . Convert to $\text{cm}^2$ .

3.2 \times 10\,000 = 32\,000\text{ cm}^2

Common mistake

multiplying an area in $\text{m}^2$ by $100$ instead of $100^2$ . Always ask "is this a length, an area, or a volume?" first, then square or cube the basic factor accordingly — a quick sanity check is that converting to a smaller unit should always make the number bigger, whatever power is used.

04 — Explanation

How do I convert between volume and capacity units?

Cube the length conversion factor:

1\text{ m}^3 = 100^3\text{ cm}^3 = 1\,000\,000\text{ cm}^3 \qquad 1\text{ litre} = 1000\text{ cm}^3 = 1000\text{ ml}

Worked example. A tank holds $2\,500\,000\text{ cm}^3$ of water. Convert to litres.

2\,500\,000 \div 1000 = 2500 \text{ litres}

05 — Explanation

How are volume and capacity connected?

Capacity measures how much liquid a container can hold, and volume measures the 3D space something occupies — for a container being filled with liquid, the two describe the same physical space, which is why $1\text{ cm}^3$ of space holds exactly $1\text{ ml}$ of liquid.

Worked example. A cuboid fish tank measures $50\text{ cm}\times30\text{ cm}\times40\text{ cm}$ . Find its capacity in litres.

V = 50\times30\times40 = 60\,000\text{ cm}^3 \qquad 60\,000\div1000 = 60\text{ litres}

Exam tip

when a question mixes units within the same shape — for example, giving one side in metres and another in centimetres — convert every measurement into the same unit before multiplying, never partway through the calculation.

06 — Explanation

How do I convert between units of mass?

Mass conversions follow the same pattern as length, since mass is a 1D measurement of "how much," not built from multiplied lengths:

1000\text{ g} = 1\text{ kg} \qquad 1000\text{ kg} = 1\text{ tonne}

Worked example. A recipe requires $750\text{ g}$ of flour. A shop only sells flour in $1.5\text{ kg}$ bags. How many grams of flour would be left over after using one bag for the recipe?

1.5\text{ kg} = 1.5\times1000 = 1500\text{ g} \qquad 1500 - 750 = 750\text{ g left over}

Exam tip

whenever a question gives quantities in mixed units (grams and kilograms, or millilitres and litres), convert everything to the smaller unit first — it avoids decimals in the working and makes the final subtraction or addition much easier to check.