Practice Maths

Units of Measurement and Conversions

← Back to Shape and Measurement›General Maths

Key Terms

The metric system uses base-10 prefixes: kilo- (1000), centi- (1/100), milli- (1/1000).
To convert to a smaller unit, multiply. To convert to a larger unit, divide.
Area conversions: multiply or divide by the square of the linear conversion factor.
Volume conversions: multiply or divide by the cube of the linear conversion factor.
Capacity
and volume link: 1 mL = 1 cm³, 1 L = 1000 cm³ = 1000 mL, 1 kL = 1000 L = 1 m³.
Mass
1 kg = 1000 g, 1 t = 1000 kg, 1 g = 1000 mg.
Conversion Factors — Length
• 1 cm = 10 mm  |  1 m = 100 cm = 1 000 mm  |  1 km = 1 000 m

Conversion Factors — Area
• 1 cm² = 100 mm²  |  1 m² = 10 000 cm²  |  1 ha = 10 000 m²  |  1 km² = 1 000 000 m² = 100 ha

Conversion Factors — Volume & Capacity
• 1 cm³ = 1000 mm³  |  1 m³ = 1 000 000 cm³  |  1 mL = 1 cm³  |  1 L = 1000 mL  |  1 kL = 1000 L = 1 m³

Conversion Factors — Mass
• 1 g = 1000 mg  |  1 kg = 1000 g  |  1 t = 1000 kg

Length Conversion Ladder — moving down multiplies, moving up divides.

km kilometres m metres cm centimetres mm millimetres ×1000 ×100 ×10 ÷1000 ÷100 ÷10 ↓ multiply (to smaller unit) ↑ divide (to larger unit)
Area & Volume Conversions — the squaring/cubing rule
Since 1 m = 100 cm, then 1 m² = 100² cm² = 10 000 cm².
Since 1 m = 100 cm, then 1 m³ = 100³ cm³ = 1 000 000 cm³.
Rule: if the linear factor is k, the area factor is k² and the volume factor is k³.
Hot Tip Always write the unit at every step of your working. This prevents multiplying when you should divide (and vice versa). Ask yourself: “Am I going to a smaller or larger unit?” Smaller unit → more of them → multiply. Larger unit → fewer of them → divide.

Worked Example 1 — Length conversion

Convert 3.45 km to metres.

km → m: multiply by 1000

3.45 km = 3.45 × 1000 = 3 450 m

Worked Example 2 — Area conversion

Convert 5.2 m² to cm².

1 m = 100 cm, so 1 m² = 100² cm² = 10 000 cm²

5.2 m² = 5.2 × 10 000 = 52 000 cm²

Worked Example 3 — Volume and capacity

A container holds 2.4 L. Express this in cm³ and mL.

1 L = 1000 mL = 1000 cm³

2.4 L = 2.4 × 1000 = 2 400 mL = 2 400 cm³

Worked Example 4 — Mixed units (real-world)

A paddock measures 450 m × 320 m. Express its area in hectares.

Area = 450 × 320 = 144 000 m²

1 ha = 10 000 m², so 144 000 ÷ 10 000 = 14.4 ha

The Metric System: A Logical Design

The metric system was deliberately designed around powers of 10, making it the universal language of science and engineering. Every prefix represents a multiplication by a power of 10 from the base unit. The three base units you will use most are the metre (length), gram (mass), and litre (capacity). A memory trick for the prefixes in order: King Henry Died By Drinking Chocolate Milk — kilo, hecto, deca, base, deci, centi, milli. Each step along that ladder multiplies or divides by 10.

Key prefix values: kilo = 1 000, centi = 0.01, milli = 0.001, mega = 1 000 000. Knowing why the system works this way means you never have to memorise isolated facts — one rule covers everything.

Converting Length

Converting between length units is a single multiply or divide. Going to a smaller unit (e.g. m → cm) means the number gets bigger, so you multiply. Going to a larger unit (e.g. mm → m) means the number gets smaller, so you divide. Write the units in your working and cancel them systematically so you can check you are going the right direction.

Example: Convert 4.7 km to metres.
1 km = 1000 m → going to a smaller unit → multiply.
4.7 × 1000 = 4700 m

Area Conversions: The Squaring Trap

This is one of the most common exam errors. Students convert 1 m² to cm² and write 100 cm² — but that is wrong. A square metre is a square with sides 1 m = 100 cm each. Its area is 100 cm × 100 cm = 10 000 cm². You must square the linear conversion factor. In general: if 1 unit⊂A = k unit⊂B, then 1 unit⊂A² = k² unit⊂B².

Common area conversions to know: 1 m² = 10 000 cm²; 1 km² = 1 000 000 m²; 1 ha = 10 000 m².

Example: Convert 3.5 m² to cm².
Linear factor: 1 m = 100 cm → Area factor: 100² = 10 000.
3.5 × 10 000 = 35 000 cm²

Volume Conversions: The Cubing Trap

The same logic applies but now we have three dimensions, each converting. So 1 m³ = (100 cm)³ = 1 000 000 cm³. You cube the linear factor. For the litre–volume link: 1 mL = 1 cm³, 1 L = 1000 cm³, and 1 kL = 1 m³. These capacity conversions appear constantly in practical questions.

Example: Convert 0.004 m³ to cm³.
Linear factor: 1 m = 100 cm → Volume factor: 100³ = 1 000 000.
0.004 × 1 000 000 = 4000 cm³

Time and Other Unit Conversions

Time is not metric, so conversions require care: 60 seconds = 1 minute; 60 minutes = 1 hour; 24 hours = 1 day. For mass: 1 kg = 1000 g; 1 tonne = 1000 kg. Always set up your conversion as a fraction so units cancel correctly. For example, to convert 3.5 hours to seconds: 3.5 h × (60 min/h) × (60 s/min) = 12 600 s.

Key Point: For area conversions, square the linear conversion factor. For volume conversions, cube it. Never just multiply by 100 or 1000 without thinking about the dimension of the quantity.
Exam Technique: Always write units in every line of working and cancel them as fractions. If your units don’t cancel correctly, your conversion is wrong. This habit also earns method marks even if you make an arithmetic error.

Mastery Practice

See Answers ➔
  1. Fluency

    Convert each length measurement.

    1. (a) 4.7 km to metres
    2. (b) 850 cm to metres
    3. (c) 3 200 mm to metres
    4. (d) 0.068 km to centimetres
  2. Fluency

    Convert each area measurement.

    1. (a) 3.5 m² to cm²
    2. (b) 48 000 cm² to m²
    3. (c) 2.4 ha to m²
    4. (d) 650 000 m² to km²
  3. Fluency

    Convert each volume or capacity measurement.

    1. (a) 4 500 cm³ to mL
    2. (b) 8.6 L to cm³
    3. (c) 2.3 m³ to kL
    4. (d) 750 mL to litres
  4. Fluency

    Convert each mass measurement.

    1. (a) 4 200 g to kg
    2. (b) 0.85 t to kg
    3. (c) 3 250 mg to g
    4. (d) 5.7 kg to grams
  5. Fluency

    Complete the conversion table for volume and capacity.

    Express each in the units indicated.
    1. (a) 1.5 m³ to cm³
    2. (b) 250 000 mm³ to cm³
    3. (c) 4 800 L to kL
    4. (d) 0.025 m³ to mL
  6. Understanding

    Multi-step conversions. Express the answer in the units indicated.

    1. (a) A rectangular garden is 12 m wide and 8 m long. Calculate the area in cm².
    2. (b) A swimming pool holds 450 kL. Express this volume in m³ and in litres.
    3. (c) A driveway is 15 m long and 3.5 m wide. A layer of gravel 0.15 m deep is laid. Find the volume in m³ and in litres.
  7. Understanding

    Area conversions in context.

    1. (a) A farm covers 340 ha. Express this in km² and in m².
    2. (b) A council reserve has an area of 0.42 km². A new road removes a strip 400 m long and 20 m wide. Find the remaining area in hectares.
    3. (c) A builder quotes to tile a room 6.4 m × 4.8 m. Each tile covers 400 cm². How many tiles are needed?
  8. Understanding

    Capacity and mass problems.

    1. (a) A water tank holds 12 500 L. Express this in kL and in m³.
    2. (b) Medicine is dispensed at 5 mL per dose. A 1.2 L bottle contains how many doses?
    3. (c) A bag of fertiliser weighs 22.5 kg. How many bags are needed to make up 1 tonne? What total mass in grams does this represent?
  9. Problem Solving

    Scale model and conversion.

    Challenge. A model of a building is made at a scale of 1 : 50.
    1. (a) The model is 32 cm tall. How tall is the actual building in metres?
    2. (b) The model has a floor area of 480 cm². What is the actual floor area in m²?
    3. (c) If the actual building's volume is 8 750 m³, what is the model's volume in cm³?
    4. (d) The model is painted with a coat 1 mm thick. Express the equivalent thickness on the real building in centimetres.
  10. Problem Solving

    Mixed real-world problem.

    Challenge. A farmer needs to irrigate a rectangular paddock 1.2 km long and 800 m wide to a depth of 50 mm of water.
    1. (a) Find the length and width of the paddock in metres.
    2. (b) Calculate the area of the paddock in hectares.
    3. (c) Calculate the volume of water required in m³.
    4. (d) Express this volume in megalitres (1 ML = 1 000 000 L = 1 000 m³).