← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 5 Mensuration · Chapter 26 · pp. 346–350

Measures

The metric system measures everything in powers of 10, so changing units is always a multiplication or a division by 10, 100 or 1000. Length is the easy part. For area the factor is squared, and for volume it is cubed.

Learning objectives

  1. Know the metric units of length, mass and capacity and choose a sensible one.
  2. Convert between units of length, mass and capacity.
  3. Convert between units of area and of volume.

Book exercises: 26.1 to 26.5 and Student assessments 1 and 2 (pp. 346–350).

The metric units

QuantityCommon units (largest first)
Lengthkilometre (km), metre (m), centimetre (cm), millimetre (mm)
Masstonne (t), kilogram (kg), gram (g), milligram (mg)
Capacitylitre (L or l), millilitre (ml)
kilo- (k) base unit centi- (c) milli- (m) × 1000 m, g, litre ÷ 100 ÷ 1000 1 km = 1000 m 1 m 1 cm = ¹⁄₁₀₀ m 1 mm = ¹⁄₁₀₀₀ m Greek: thousand no prefix Latin centum: 100 Latin mille: 1000
The prefix tells you the size. The same prefixes work for every quantity: a kilogram is 1000 grams, a millilitre is one thousandth of a litre.

Choosing a sensible unit

Before converting, get a feel for the sizes. These benchmarks help with estimating:

UnitAbout the size of…
1 mmthe thickness of a bank card
1 cmthe width of your little finger
1 mone big stride, or the height of a door handle
1 kmabout 12 minutes' walk
1 ga paper clip
1 kga bag of sugar
1 ta small car
1 ml / 1 litrea few drops / a large bottle of water

📘 Example 1 · Which unit for the length of a classroom?

  1. It is a length, so choose from mm, cm, m or km.
  2. A classroom is several strides long, so a stride-sized unit fits.
  3. Answer: metres (m). A classroom is about 8 m long.

✏️ Try it

  1. How many milligrams are in one gram?
  2. Which unit would you use for the length of a football pitch?
  3. Complete: one centimetre is one … of a metre.
Answers

1. 1000 mg · 2. metres (m) · 3. hundredth

Next: converting length →

Converting length

🔑 Key facts

  • 1 km = 1000 m, so 1 m = km.
  • 1 m = 100 cm, so 1 cm = m.
  • 1 m = 1000 mm, so 1 mm = m.
  • 1 cm = 10 mm, so 1 mm = cm.
  • Bigger unit → smaller unit: multiply (you need more of the small units). Smaller → bigger: divide.

📕 Book correction: the book prints 1 mm = cm. Since 1 cm = 10 mm, it should be 1 mm = cm.

km × 1000 ÷ 1000 m × 100 ÷ 100 cm × 10 ÷ 10 mm Down a step (smaller unit): multiply Up a step (bigger unit): divide
The length staircase. Walking down the stairs you multiply by the number on each step; walking up you divide. km to cm crosses two steps: × 1000 then × 100.

📘 Example 2 · Change 5.8 km into m

  1. 1 km = 1000 m.
  2. km to m is bigger to smaller, so multiply: 5.8 × 1000.
  3. Answer: 5.8 km = 5800 m.

📘 Example 3 · Change 4700 mm to m

  1. 1 m = 1000 mm.
  2. mm to m is smaller to bigger, so divide: 4700 ÷ 1000.
  3. Answer: 4700 mm = 4.7 m.

📘 Example 4 · Convert 2.3 km into cm

  1. Go down one step at a time. km to m: 2.3 × 1000 = 2300 m.
  2. m to cm: 2300 × 100 = 230 000 cm.
  3. Answer: 2.3 km = 230 000 cm.
  4. Check: one combined factor, 1000 × 100 = 100 000, and 2.3 × 100 000 = 230 000 ✓

📘 Example 5 · Convert 75 mm to m

  1. 1 m = 1000 mm.
  2. Smaller to bigger, so divide: 75 ÷ 1000.
  3. Answer: 75 mm = 0.075 m.
  4. Check: a metre is far longer than 75 mm, so the answer should be well under 1 ✓

✏️ Try it

  1. Convert 7.3 km to m.
  2. Convert 45 mm to cm.
  3. Convert 0.6 km to cm.
Answers

1. 7300 m · 2. 4.5 cm · 3. 0.6 × 1000 × 100 = 60 000 cm

Next: mass and capacity →

Mass and capacity

FactSo…
1 tonne = 1000 kg1 kg = tonne
1 kg = 1000 g1 g = kg
1 g = 1000 mg1 mg = g
1 litre = 1000 ml1 ml = litre
Mass: every step is 1000 tonne (t) ×1000 → ← ÷1000 kilogram (kg) ×1000 → ← ÷1000 gram (g) ×1000 → ← ÷1000 milligram (mg) Capacity: one step of 1000 litre millilitre (ml) ×1000 → / ← ÷1000 Useful: 1 ml = 1 cm³
Mass and capacity both move in steps of 1000. To the right (smaller unit) multiply; to the left (bigger unit) divide.

📘 Example 6 · Convert 8300 kg to tonnes

  1. 1000 kg = 1 tonne.
  2. kg to tonnes is smaller to bigger, so divide: 8300 ÷ 1000.
  3. Answer: 8300 kg = 8.3 tonnes.

📘 Example 7 · Convert 2.5 g to mg

  1. 1 g = 1000 mg.
  2. Bigger to smaller, so multiply: 2.5 × 1000.
  3. Answer: 2.5 g = 2500 mg.

📘 Example 8 · Convert 0.35 kg to mg

  1. kg to g: 0.35 × 1000 = 350 g.
  2. g to mg: 350 × 1000 = 350 000 mg.
  3. Answer: 0.35 kg = 350 000 mg.

📘 Example 9 · Work out 2.4 litres + 750 ml. Give the total in ml, then in litres.

  1. You can only add amounts in the same unit. Change litres to ml: 2.4 × 1000 = 2400 ml.
  2. Add: 2400 + 750 = 3150 ml.
  3. Change back to litres: 3150 ÷ 1000 = 3.15 litres.
  4. Answer: 3150 ml = 3.15 litres.

✏️ Try it

  1. Convert 4.6 kg to g.
  2. Convert 650 kg to tonnes.
  3. Work out 1.5 litres + 800 ml, in litres.
Answers

1. 4600 g · 2. 0.65 tonnes · 3. 1500 + 800 = 2300 ml = 2.3 litres

Next: area conversions →

Area conversions

Area units are squares, so the length factor gets used twice, once for each side.

1 m 1 m Area = 1 m² = 100 cm 100 cm 100 × 100 = 10 000 cm²
The same square measured two ways. Each small square in the grid is 10 cm × 10 cm = 100 cm², and there are 100 of them: 10 000 cm² in 1 m².
Length factSquare itArea fact
1 cm = 10 mm10 × 101 cm² = 100 mm²
1 m = 100 cm100 × 1001 m² = 10 000 cm²
1 m = 1000 mm1000 × 10001 m² = 1 000 000 mm²
1 km = 1000 m1000 × 10001 km² = 1 000 000 m²

💡 The most common mistake: 1 m² = 100 cm². It is not. Draw the square and you will see 100 cm along each side.

📘 Example 10 · Convert 3.5 m² to cm²

  1. Length: 1 m = 100 cm.
  2. Square it: 1 m² = 100 × 100 = 10 000 cm².
  3. Bigger to smaller unit, so multiply: 3.5 × 10 000.
  4. Answer: 3.5 m² = 35 000 cm².

📘 Example 11 · Convert 45 000 mm² to cm²

  1. Length: 1 cm = 10 mm.
  2. Square it: 1 cm² = 10 × 10 = 100 mm².
  3. Smaller to bigger unit, so divide: 45 000 ÷ 100.
  4. Answer: 45 000 mm² = 450 cm².

📘 Example 12 · Convert 2 km² to m²

  1. Length: 1 km = 1000 m.
  2. Square it: 1 km² = 1000 × 1000 = 1 000 000 m².
  3. Multiply: 2 × 1 000 000.
  4. Answer: 2 km² = 2 000 000 m².

✏️ Try it

  1. Convert 6 m² to cm².
  2. Convert 250 mm² to cm².
  3. Convert 0.7 km² to m².
Answers

1. 60 000 cm² · 2. 250 ÷ 100 = 2.5 cm² · 3. 700 000 m²

Next: volume conversions →

Volume conversions

Volume units are cubes, so the length factor gets used three times: length, width and height.

100 cm 100 cm 100 cm Volume = 1 m³ Length: × 100 Area: × 100² = × 10 000 Volume: × 100³ = × 1 000 000
A 1 m cube is 100 cm along every edge, so its volume is 100 × 100 × 100 = 1 000 000 cm³. Length factor, then square it for area, cube it for volume.
Length factCube itVolume fact
1 cm = 10 mm10 × 10 × 101 cm³ = 1000 mm³
1 m = 100 cm100 × 100 × 1001 m³ = 1 000 000 cm³
1 km = 1000 m1000³1 km³ = 1 000 000 000 m³

📘 Example 13 · Convert 0.4 m³ to cm³

  1. Length: 1 m = 100 cm.
  2. Cube it: 1 m³ = 100 × 100 × 100 = 1 000 000 cm³.
  3. Bigger to smaller unit, so multiply: 0.4 × 1 000 000.
  4. Answer: 0.4 m³ = 400 000 cm³.

📘 Example 14 · Convert 7500 mm³ to cm³

  1. Length: 1 cm = 10 mm.
  2. Cube it: 1 cm³ = 10 × 10 × 10 = 1000 mm³.
  3. Smaller to bigger unit, so divide: 7500 ÷ 1000.
  4. Answer: 7500 mm³ = 7.5 cm³.

🧪 Try it: metric unit converter

    ✏️ Try it

    1. Convert 3 m³ to cm³.
    2. Convert 4.5 cm³ to mm³.
    3. Convert 250 000 cm³ to m³.
    Answers

    1. 3 000 000 cm³ · 2. 4500 mm³ · 3. 250 000 ÷ 1 000 000 = 0.25 m³

    Next: summary and quiz →

    Chapter summary

    • Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm (so 1 m = 1000 mm).
    • Mass: 1 t = 1000 kg, 1 kg = 1000 g, 1 g = 1000 mg. Capacity: 1 litre = 1000 ml.
    • Bigger unit → smaller unit: multiply. Smaller → bigger: divide.
    • Add or subtract amounts only once they are in the same unit.
    • Area: square the length factor (1 m² = 10 000 cm², 1 cm² = 100 mm², 1 km² = 1 000 000 m²).
    • Volume: cube the length factor (1 m³ = 1 000 000 cm³, 1 cm³ = 1000 mm³).

    📄 Practice questions

    The book's Exercises 26.1 to 26.5 and Student assessments 1 and 2 are in the printable assignment for this chapter.

    Mini-quiz

    1. How many millimetres are there in 1 metre?
    2. Convert 3.6 km to metres.
    3. Convert 450 cm to metres.
    4. Which unit is best for the mass of a paper clip?
    5. Convert 2.4 tonnes to kilograms.
    6. How many grams is 750 mg?
    7. Work out 1.2 litres + 350 ml. Give the total in ml.
    8. How many cm² are there in 1 m²?
    9. Convert 4 cm² to mm².
    10. How many cm³ are there in 1 m³?
    11. Convert 5000 mm³ to cm³.
    12. From cm to mm the length factor is 10. What is the factor for volume?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 26 (Measures). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.