Units of measure

1. Overview

This revision note covers units of measure, focusing on the metric system as used in IGCSE Mathematics (0580). A crucial skill is converting between units of length, mass, area, volume, and capacity. Before performing any calculations in geometry or other topics, ensure all measurements are expressed in the same units. This note provides the essential conversion factors, worked examples, and common mistakes to avoid, helping you master this fundamental topic.

Key Definitions

• Metric System: A decimal-based system of measurement used internationally (e.g., metres, grams, litres).
• Length: The measurement of something from end to end (mm, cm, m, km).
• Mass: The amount of matter in an object (mg, g, kg, t).
• Capacity: The amount of liquid a container can hold (ml, cl, l).
• Volume: The amount of 3D space an object occupies (mm³, cm³, m³).
• Conversion Factor: The numerical value used to multiply or divide a quantity to change its units.

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Core Content

A. Linear Units (Length)

Standard units: Millimetres (mm), Centimetres (cm), Metres (m), Kilometres (km).

The Conversion Rules:

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

Worked example 1 — Kilometres to Metres

Question: Convert 4.5 kilometres into metres.

1. Identify the conversion factor: $1 \text{ km} = 1000 \text{ m}$.
2. We are going from a larger unit (km) to a smaller unit (m), so we multiply.
3. Calculation: $4.5 \times 1000 = 4500$.

Answer: $4500 \text{ m}$

Worked example 2 — Millimetres to Metres

Question: Convert 3500 millimetres into metres.

1. Identify the conversion factors: $10 \text{ mm} = 1 \text{ cm}$ and $100 \text{ cm} = 1 \text{ m}$. Therefore, $1000 \text{ mm} = 1 \text{ m}$.
2. We are going from a smaller unit (mm) to a larger unit (m), so we divide.
3. Calculation: $3500 \div 1000 = 3.5$.

Answer: $3.5 \text{ m}$

B. Mass and Capacity

Mass:

• $1000 \text{ mg} = 1 \text{ g}$
• $1000 \text{ g} = 1 \text{ kg}$
• $1000 \text{ kg} = 1 \text{ tonne (t)}$

Capacity:

• $10 \text{ ml} = 1 \text{ cl}$
• $1000 \text{ ml} = 1 \text{ l}$
• $100 \text{ cl} = 1 \text{ l}$

Special Relationship: $1 \text{ cm}^3 = 1 \text{ ml}$

Worked example 3 — Litres to Millilitres

Question: A bottle contains 0.75 litres of water. How many millilitres is this?

1. Identify the conversion factor: $1 \text{ l} = 1000 \text{ ml}$.
2. We are going from a larger unit (litres) to a smaller unit (millilitres), so we multiply.
3. Calculation: $0.75 \times 1000 = 750$.

Answer: $750 \text{ ml}$

Worked example 4 — Kilograms to Grams

Question: A bag of sugar weighs 2.3 kilograms. Convert this mass to grams.

1. Identify the conversion factor: $1 \text{ kg} = 1000 \text{ g}$.
2. We are going from a larger unit (kilograms) to a smaller unit (grams), so we multiply.
3. Calculation: $2.3 \times 1000 = 2300$.

Answer: $2300 \text{ g}$

C. Converting Area and Volume Units

This is where many students lose marks. You cannot use linear conversion factors for area or volume.

• For Area: Square the linear conversion factor.
  • Since $1 \text{ cm} = 10 \text{ mm}$, then $1 \text{ cm}^2 = 10^2 \text{ mm}^2 = 100 \text{ mm}^2$.
• For Volume: Cube the linear conversion factor.
  • Since $1 \text{ m} = 100 \text{ cm}$, then $1 \text{ m}^3 = 100^3 \text{ cm}^3 = 1,000,000 \text{ cm}^3$.

Worked example 5 — Metres Squared to Centimetres Squared

Question: Convert $3 \text{ m}^2$ into $\text{cm}^2$.

1. Linear factor: $1 \text{ m} = 100 \text{ cm}$.
2. Area factor: $100^2 = 10,000$.
3. Calculation: $3 \times 10,000 = 30,000$.

Answer: $30,000 \text{ cm}^2$

Worked example 6 — Centimetres Cubed to Millimetres Cubed

Question: Convert $2.5 \text{ cm}^3$ into $\text{mm}^3$.

1. Linear factor: $1 \text{ cm} = 10 \text{ mm}$.
2. Volume factor: $10^3 = 1000$.
3. Calculation: $2.5 \times 1000 = 2500$.

Answer: $2500 \text{ mm}^3$

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Extended Content (Extended Only)

While there are no specific additional objectives for Topic 5.1 in the Extended curriculum, you're expected to apply these unit conversions in more complex problems. This often involves compound shapes, density calculations, and pressure problems. A common application is calculating density, where $Density = \frac{Mass}{Volume}$. Ensure that the mass and volume are in consistent units (e.g., kg/m³ or g/cm³) before performing the division. Another area is working with compound shapes, where you might need to convert all lengths to the same unit before calculating areas or volumes. For example, a prism might have its length in metres and its cross-section dimensions in centimetres; you'd need to convert everything to either metres or centimetres before finding the volume.

Worked example 7 — Density Calculation

Question: A metal block has a mass of 5 kg and a volume of 2000 cm³. Calculate the density of the metal in g/cm³.

1. Identify the required units: We need the density in g/cm³, but the mass is in kg.
2. Convert the mass from kg to g: $1 \text{ kg} = 1000 \text{ g}$, so $5 \text{ kg} = 5 \times 1000 = 5000 \text{ g}$.
3. Apply the density formula: $Density = \frac{Mass}{Volume} = \frac{5000 \text{ g}}{2000 \text{ cm}^3}$.
4. Calculation: $Density = 2.5 \text{ g/cm}^3$.

Answer: $2.5 \text{ g/cm}^3$

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Key Equations

Note: None of these conversion factors are provided on the IGCSE formula sheet. You must memorise them.

Dimension	Conversion Factors
Length	$1 \text{ km} = 10^3 \text{ m}$
Area	$1 \text{ km}^2 = (10^3)^2 \text{ m}^2$
Volume	$1 \text{ km}^3 = (10^3)^3 \text{ m}^3$

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Common Mistakes to Avoid

• ❌ Wrong: Converting $7 \text{ m}^2$ to $\text{cm}^2$ by multiplying by 100 (Result: $700 \text{ cm}^2$).
• ✓ Right: Square the factor. $7 \times 100^2 = 7 \times 10000 = 70,000 \text{ cm}^2$.
• ❌ Wrong: A question provides the radius of a circle in cm, but asks for the area in mm². Calculating the area in cm² and then converting the final answer.
• ✓ Right: Convert the radius from cm to mm first, and then calculate the area in mm² directly. This avoids errors with squaring the units.
• ❌ Wrong: Confusing $1000 \text{ ml} = 1 \text{ l}$ with $100 \text{ ml} = 1 \text{ l}$.
• ✓ Right: Double-check the conversion factors, especially for capacity. Use the mnemonic "King Henry Died Monday Drinking Chocolate Milk" to remember the metric prefixes (kilo, hecto, deca, base, deci, centi, milli).
• ❌ Wrong: Forgetting to convert units in density problems, leading to incorrect density values.
• ✓ Right: Always ensure mass and volume are in compatible units (e.g., g and cm³, or kg and m³) before calculating density.

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Exam Tips

• Command Words: Look for "Convert" (simple change) or "Calculate... giving your answer in [unit]" (multi-step). Circle the units specified in the question before you start working.
• Calculator Tip: For area and volume conversions, use the $x^2$ or $x^3$ button on your calculator to avoid manual calculation of the squared or cubed conversion factor. For example, to convert from m³ to cm³, enter 100 (cm/m) then press the $x^3$ button to get 1,000,000.
• Real-world Context: Be prepared for "fencing" (length), "painting a wall" (area), or "filling a tank" (capacity/volume) questions. Visualise the scenario to help you determine the appropriate units and conversions.
• Check for Sensibility: If you calculate that a small swimming pool holds $50 \text{ ml}$ instead of $50 \text{ m}^3$, you have likely made an error in your conversion. Does the answer make sense in the real world?
• The "Double Check": Always look at the answer line. If it says __________ mm, and your working is in cm, you must do one last conversion or you will lose the final accuracy mark. Do this check before you run out of time.