Cambridge IGCSE Mathematics 0580: Units of measure, area and perimeter – Study Guide

This study guide teaches two subtopics of Topic 5 Mensuration in the Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027: 5.1 Units of measure and 5.2 Area and perimeter. The Core statements (C5.1, C5.2) and the Extended statements (E5.1, E5.2) say the same thing, so everything on this page applies to both tiers. There is no Extended-only content in these two subtopics.

Core candidates take Papers 1 (non-calculator) and 3 (calculator); Extended candidates take Papers 2 (non-calculator) and 4 (calculator). Both subtopics can appear on all four papers, so practise by hand as well as with a calculator.

Use this page with the mathematics course hub and the printable 0580 checklist. The whole of Topic 5 is summarised in the Mensuration study guide and the Mensuration revision notes, and you can test yourself with the Mensuration practice questions. This page goes deeper into 5.1 and 5.2 only.

What these subtopics cover

0580 statement	What you must be able to do	Tier
5.1	Use metric units of mass, length, area, volume and capacity in practical situations	Core and Extended
5.1	Convert quantities into larger or smaller units, including between units of area (e.g. cm² ↔ m²) and between units of volume and capacity (e.g. m³ ↔ litres)	Core and Extended
5.2	Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium	Core and Extended

The units the syllabus lists are:

• length: mm, cm, m, km
• area: mm², cm², m², km²
• volume: mm³, cm³, m³
• capacity: ml, l
• mass: g, kg

For 5.2 the syllabus says that, except for the area of a triangle, formulas are not given. The triangle formula A = ½bh is in the List of formulas on page 2 of every paper. You must learn the rectangle, parallelogram and trapezium formulas yourself.

Circles, solids and compound shapes (5.3 to 5.5) are in the Mensuration pages linked above.

5.1 Units of measure

The basic facts

Every conversion is built from these facts.

length    1 cm = 10 mm      1 m = 100 cm      1 km = 1000 m
mass      1 kg = 1000 g
capacity  1 l  = 1000 ml
volume to capacity   1 cm³ = 1 ml   so   1000 cm³ = 1 l

Which way to go: multiply or divide?

• Going to a smaller unit (km → m, kg → g): there will be more of them, so multiply.
• Going to a larger unit (mm → cm, ml → l): there will be fewer of them, so divide.

If 3 m becomes 0.03 cm, you went the wrong way.

Worked example 1: length, mass and capacity

Convert (a) 2.75 km to metres and to centimetres, (b) 4580 g to kilograms, (c) 650 ml to litres.

(a) km -> m is to a smaller unit: × 1000      2.75 × 1000 = 2750 m
    m -> cm is to a smaller unit: × 100       2750 × 100 = 275 000 cm
(b) g -> kg is to a larger unit: ÷ 1000       4580 ÷ 1000 = 4.58 kg
(c) ml -> l is to a larger unit: ÷ 1000       650 ÷ 1000 = 0.65 l

Area units: square the factor

A square of side 1 m is also a square of side 100 cm. Its area is 1 m², and it is also 100 × 100 = 10 000 cm². So:

1 cm² = 10 × 10       = 100 mm²
1 m²  = 100 × 100     = 10 000 cm²
1 km² = 1000 × 1000   = 1 000 000 m²

The rule: find the length factor, then square it. Do not use the length factor on its own. 1 m² is not 100 cm².

Worked example 2: area conversions

Convert (a) 0.36 m² to cm², (b) 52 000 mm² to cm², (c) 4.2 km² to m².

(a) m -> cm length factor 100, area factor 100² = 10 000
    smaller unit, so multiply: 0.36 × 10 000 = 3600 cm²
(b) mm -> cm length factor 10, area factor 10² = 100
    larger unit, so divide: 52 000 ÷ 100 = 520 cm²
(c) km -> m length factor 1000, area factor 1000² = 1 000 000
    smaller unit, so multiply: 4.2 × 1 000 000 = 4 200 000 m²

Check (a): a 0.6 m square has area 0.36 m², and 60 × 60 = 3600 cm².

Volume units: cube the factor

A cube of side 1 cm is also a cube of side 10 mm, so its volume is 10 × 10 × 10 = 1000 mm³.

1 cm³ = 10³  = 1000 mm³
1 m³  = 100³ = 1 000 000 cm³

So 2.5 cm³ = 2.5 × 1000 = 2500 mm³, and 750 000 cm³ = 750 000 ÷ 1 000 000 = 0.75 m³.

Volume and capacity

Capacity is how much a container holds. The link is 1 cm³ = 1 ml, so:

1000 cm³ = 1000 ml = 1 l
1 m³ = 1 000 000 cm³ = 1 000 000 ml = 1000 l

So 1 m³ = 1000 litres, the syllabus’s own example (m³ ↔ litres).

Worked example 3: a tank in mixed units

A tank is a cuboid measuring 1.2 m by 80 cm by 50 cm. Find its capacity in litres.

Convert to one unit before multiplying.

Method 1: work in cm
1.2 m = 120 cm
volume = 120 × 80 × 50 = 480 000 cm³
capacity = 480 000 ÷ 1000 = 480 litres

Method 2: work in m
80 cm = 0.8 m,  50 cm = 0.5 m
volume = 1.2 × 0.8 × 0.5 = 0.48 m³
capacity = 0.48 × 1000 = 480 litres

Multiplying the dimensions as given (1.2 × 80 × 50) gives a number with no meaningful unit.

Units in practical situations

For “practical situations”:

1. Read every quantity and note its unit.
2. Convert so that everything is in one consistent unit.
3. Calculate.
4. Convert the answer to the unit the question asks for, and write the unit.

Short example. A medicine bottle holds 0.25 l. One dose is 5 ml. How many doses are in the bottle? Convert first: 0.25 l = 250 ml. Then 250 ÷ 5 = 50 doses.

Worked example 4: tiling a floor

A rectangular floor measures 3.6 m by 2.4 m. It is covered with square tiles of side 30 cm, with no gaps. How many tiles are needed?

Convert the floor to cm: 360 cm by 240 cm
tiles along the length: 360 ÷ 30 = 12
tiles along the width:  240 ÷ 30 = 8
number of tiles = 12 × 8 = 96

Check by area: 86 400 cm² ÷ 900 cm² = 96.

5.2 Area and perimeter

Perimeter

The perimeter is the total distance around the outside of a shape. Add the lengths of all the sides. Its unit is a length unit, never a squared unit.

For a rectangle with length l and width w:

perimeter = l + w + l + w = 2(l + w)

For other shapes, add the sides. A missing side in a right-angled triangle needs Pythagoras’ theorem (see the Pythagoras’ theorem study guide).

Area formulas

Area is the amount of flat surface inside a shape, measured in squared units.

Shape	Area	Given in the List of formulas?
Rectangle	A = lw	No, learn it
Triangle	A = ½bh	Yes
Parallelogram	A = bh	No, learn it
Trapezium	A = ½(a + b)h	No, learn it

In the trapezium formula, a and b are the two parallel sides and h is the perpendicular distance between them.

Why these work. Moving a triangle from one end of a parallelogram to the other makes a rectangle, so A = bh. Two identical trapeziums, one upside down, make a parallelogram of base (a + b) and height h, so one trapezium is ½(a + b)h.

Perpendicular height

In every formula, h is the perpendicular height: it meets the base at a right angle. Use slanted sides for the perimeter and the perpendicular height for the area.

Worked example 5: a parallelogram

A parallelogram has base 12 cm and slanted sides of 7.5 cm. The perpendicular height from the base is 6 cm. Find (a) its area, (b) its perimeter.

(a) A = bh = 12 × 6 = 72 cm²
    (not 12 × 7.5: 7.5 cm is a slanted side, not the height)
(b) perimeter = 12 + 7.5 + 12 + 7.5 = 39 cm

Worked example 6: a trapezium (non-calculator)

A trapezium has parallel sides of 7.4 cm and 11.6 cm, and the perpendicular distance between them is 5 cm. Find its area.

a + b = 7.4 + 11.6 = 19
A = ½ × 19 × 5
  = ½ × 95
  = 47.5 cm²

Add the parallel sides first, then halve wherever the arithmetic is easiest.

Working backwards

Given an area or perimeter, write the formula, substitute, then solve for the missing length.

Worked example 7: finding a missing length

(a) A triangle has area 42 cm² and base 12 cm. Find its perpendicular height.

42 = ½ × 12 × h
42 = 6h
h = 7 cm

(b) A trapezium has area 60 cm² and parallel sides 9 cm and 15 cm. Find the distance between the parallel sides.

60 = ½ × (9 + 15) × h
60 = 12h
h = 5 cm

(c) A rectangle has perimeter 46 cm and length 15 cm. Find its width and its area.

2(15 + w) = 46
15 + w = 23
w = 8 cm
area = 15 × 8 = 120 cm²

Worked example 8: a right-angled triangle

A right-angled triangle has shorter sides 9 cm and 12 cm. Find (a) its area, (b) its perimeter.

(a) the two shorter sides are perpendicular, so one is the base and the other the height
    A = ½ × 9 × 12 = 54 cm²
(b) hypotenuse² = 9² + 12² = 81 + 144 = 225, so hypotenuse = 15 cm
    perimeter = 9 + 12 + 15 = 36 cm

Worked example 9: using algebra

A rectangle has length (2x + 3) cm and width x cm. Its perimeter is 42 cm. Find x and the area of the rectangle.

perimeter = 2(2x + 3 + x) = 2(3x + 3) = 6x + 6
6x + 6 = 42
6x = 36
x = 6
length = 2 × 6 + 3 = 15 cm,  width = 6 cm
area = 15 × 6 = 90 cm²

Check: 2(15 + 6) = 42.

Putting 5.1 and 5.2 together

Worked example 10: fertilising a lawn (calculator)

A lawn is a trapezium. Its parallel sides are 14 m and 22 m, and the perpendicular distance between them is 9 m. Fertiliser is spread at 40 g per square metre. It is sold in 2.5 kg bags. How many bags must be bought?

area = ½ × (14 + 22) × 9 = ½ × 36 × 9 = 162 m²
fertiliser = 162 × 40 = 6480 g
convert to kg: 6480 ÷ 1000 = 6.48 kg
bags: 6.48 ÷ 2.5 = 2.592

2 bags would not be enough, so the answer is 3 bags. This is a context decision, not a rounding rule: even 2.1 would need 3 bags.

Common errors

• Converting 0.36 m² to 36 cm² or 3.6 cm² by using the length factor 100 (or 10) instead of the area factor 10 000.
• Converting cm³ to m³ by dividing by 100 instead of 1 000 000.
• Taking 1 m³ as 1 litre or 100 litres instead of 1000 litres.
• Multiplying lengths in different units, such as 1.2 m × 80 cm × 50 cm, without converting first.
• Using a slanted side instead of the perpendicular height for a parallelogram, triangle or trapezium.
• Using the two non-parallel sides of a trapezium as a and b.
• Leaving out the ½ in the triangle or trapezium formula.
• Giving a perimeter in cm² or an area in cm.
• Rounding a “how many bags/tins/tiles” answer down when that would not be enough.

Next steps

• Cover all of Topic 5 with the Mensuration study guide and condense it with the Mensuration revision notes.
• Practise with the Mensuration practice questions.
• Revise shape properties in the Geometry study guide.
• Find your gaps across the whole course with the free diagnostic: Core or Extended.

Official syllabus

Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027 (Version 3), Cambridge International. Topic 5 Mensuration, subtopics C5.1 and E5.1 Units of measure, and C5.2 and E5.2 Area and perimeter.