Welcome to Mensuration: Mastering Units of Measure!

Hello! This chapter is all about understanding the language of measurement. In Mathematics, especially in the Mensuration section (calculating areas, volumes, and perimeters), if you use the wrong units or convert them incorrectly, your final answer will be wrong—even if your main formula is perfect!

We will focus entirely on the Metric System, which is standard in IGCSE Mathematics. Learning how to move smoothly between small units (like millilitres) and large units (like kilometres) is an essential skill, both for exams and for real life!


1. The Foundation: Metric Prefixes and Conversions (Length and Mass)

The metric system is wonderful because it is based on powers of 10. This means moving between units is just multiplication or division by 10, 100, or 1000.

1.1 Key Units to Know (Length, Mass, Capacity)

The syllabus requires you to be familiar with the following basic units:

  • Length (1D): Millimetres (mm), Centimetres (cm), Metres (m), Kilometres (km).
  • Mass: Grams (g), Kilograms (kg).
  • Capacity: Millilitres (ml), Litres (l).

1.2 The Magic Number: 1000 (And the Mnemonic)

In the metric system, conversions often involve 1000.

  • \(1 \text{ km} = 1000 \text{ m}\)
  • \(1 \text{ kg} = 1000 \text{ g}\)
  • \(1 \text{ l} = 1000 \text{ ml}\)
  • \(1 \text{ m} = 100 \text{ cm}\)
  • \(1 \text{ cm} = 10 \text{ mm}\)

🔥 Memory Aid for Length Conversions:
Think of moving the decimal point:
To go from a Large unit to a Small unit (e.g., m to cm), you Multiply.
To go from a Small unit to a Large unit (e.g., cm to m), you Divide.

Example: Convert 2.5 kg to grams.
You are going from a large unit (kg) to a small unit (g). You multiply by 1000.
\(2.5 \times 1000 = 2500 \text{ g}\)

Quick Review: Length Steps

\(1 \text{ km} \xrightarrow{\times 1000} 1000 \text{ m} \xrightarrow{\times 100} 100000 \text{ cm} \xrightarrow{\times 10} 1000000 \text{ mm}\)


2. Units of Area (Squared Conversions)

When we measure Area, we are working in two dimensions (Length \(\times\) Width). Because of this, the conversion factors are now squared.

2.1 Understanding the Conversion Multipliers

If the conversion factor for length is \(x\), the conversion factor for area will be \(x^2\).

  • Metres to Centimetres:
    Length: \(1 \text{ m} = 100 \text{ cm}\)
    Area: \(1 \text{ m}^2 = 100^2 \text{ cm}^2 = 10000 \text{ cm}^2\)
  • Centimetres to Millimetres:
    Length: \(1 \text{ cm} = 10 \text{ mm}\)
    Area: \(1 \text{ cm}^2 = 10^2 \text{ mm}^2 = 100 \text{ mm}^2\)
  • Kilometres to Metres:
    Length: \(1 \text{ km} = 1000 \text{ m}\)
    Area: \(1 \text{ km}^2 = 1000^2 \text{ m}^2 = 1000000 \text{ m}^2\)

💡 Why this happens?
Imagine a square that is \(1 \text{ metre}\) by \(1 \text{ metre}\). Its area is \(1 \text{ m}^2\).
If you measure the same square in centimetres, it is \(100 \text{ cm}\) by \(100 \text{ cm}\).
Area in cm: \(100 \times 100 = 10,000 \text{ cm}^2\).
The area itself hasn't changed, only the unit size!

2.2 Step-by-Step Area Conversion

Don't worry if this seems tricky at first. Always use the two-step method:

  1. Find the basic length conversion factor (e.g., cm to m is \(\div 100\)).
  2. Square that factor for area conversion (e.g., Area conversion is \(\div 100^2\)).

Example: Convert \(500 \text{ cm}^2\) to \(\text{m}^2\).
Step 1: Length conversion is \(1 \text{ m} = 100 \text{ cm}\). So, we divide by 100.
Step 2: For area, we divide by \(100^2 = 10000\).
\(500 \div 10000 = 0.05 \text{ m}^2\)


3. Units of Volume and Capacity (Cubed Conversions)

Volume measures three dimensions (Length \(\times\) Width \(\times\) Height). Therefore, the conversion factors are now cubed.

3.1 Volume Conversions: Cubing the Factor

If the conversion factor for length is \(x\), the conversion factor for volume will be \(x^3\).

  • Metres to Centimetres:
    Length: \(1 \text{ m} = 100 \text{ cm}\)
    Volume: \(1 \text{ m}^3 = 100^3 \text{ cm}^3 = 1,000,000 \text{ cm}^3\)
  • Centimetres to Millimetres:
    Length: \(1 \text{ cm} = 10 \text{ mm}\)
    Volume: \(1 \text{ cm}^3 = 10^3 \text{ mm}^3 = 1000 \text{ mm}^3\)

3.2 The Crucial Link: Volume to Capacity

Capacity refers to how much liquid (or substance) a container can hold. The syllabus requires you to know the relationship between volume (e.g., \(\text{cm}^3\), \(\text{m}^3\)) and capacity (e.g., ml, l).

These are the two relationships you must memorize:

Key Relationship 1: Small Units
\(1 \text{ cm}^3 = 1 \text{ ml}\)
(A cube 1 cm by 1 cm by 1 cm holds 1 millilitre of liquid.)

Key Relationship 2: Large Units
\(1 \text{ m}^3 = 1000 \text{ litres}\)
(A large water tank 1 metre wide, 1 metre deep, and 1 metre high holds 1000 litres.)

Did you know? Because \(1 \text{ m}^3\) is \(1000 \text{ l}\), and \(1 \text{ l}\) is \(1000 \text{ ml}\), this means:

\(1 \text{ m}^3 = 1000 \times 1000 \text{ ml} = 1,000,000 \text{ ml}\)
And since \(1 \text{ cm}^3 = 1 \text{ ml}\), this confirms that \(1 \text{ m}^3 = 1,000,000 \text{ cm}^3\)!

3.3 Step-by-Step Volume/Capacity Conversion

Example: A swimming pool has a volume of \(5 \text{ m}^3\). How many litres does it hold?

Step 1: Identify the relationship: \(1 \text{ m}^3 = 1000 \text{ l}\).
Step 2: Calculate the total capacity.
\(5 \times 1000 = 5000 \text{ litres}\).

Example: Convert \(4500 \text{ mm}^3\) to \(\text{cm}^3\).
Step 1: Length conversion is mm to cm, so \(\div 10\).
Step 2: For volume, we divide by \(10^3 = 1000\).
\(4500 \div 1000 = 4.5 \text{ cm}^3\)


4. Working with Mass and Density

Although mass (g, kg) conversions are straightforward (always based on 1000), they often appear in problems involving Density.

Density is a rate, linking mass and volume.

\(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\)

The standard notation for density you will see in the exam is often \(g/\text{cm}^3\) (grams per cubic centimetre) or \(\text{kg}/\text{m}^3\) (kilograms per cubic metre).

To solve these problems, all your units must match the units of the given density.

⚠️ Common Mistake to Avoid

Students often forget to cube or square the conversion factor. Never simply divide area/volume by 10 or 100!
If you convert 3 m² to cm², you MUST use \(100^2\), not just 100.

Always check the power of the unit:

  • \(\text{Length} \rightarrow \text{Factor}^1\)
  • \(\text{Area} \rightarrow \text{Factor}^2\)
  • \(\text{Volume} \rightarrow \text{Factor}^3\)

Key Takeaways for Units of Measure (C6.1 / E6.1)

  • Length (1D): Conversion factor is \(x\). (\(10, 100, \text{ or } 1000\)).
  • Area (2D): Conversion factor is \(x^2\).
  • Volume (3D): Conversion factor is \(x^3\).
  • Mass: Conversions are always based on 1000 (\(\text{g} \leftrightarrow \text{kg}\)).
  • Capacity Link: Know the critical equivalencies:
    • \(1 \text{ cm}^3 = 1 \text{ ml}\)
    • \(1 \text{ m}^3 = 1000 \text{ l}\)
  • Practice Tip: Always write down the conversion factor first (e.g., "I need to convert m³ to cm³, so I multiply by \(100^3\)"), as this shows your clear method.