Area and Volume Conversions

Change area units with a squared factor, volume units with a cubed factor, and change volumes in cm³ or m³ into millilitres and litres.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 25 real past-paper questions below

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Why the Number Is Squared or Cubed

A metre is 100 cm long. A square metre is a square 100 cm long and 100 cm wide, so it holds far more than 100 square centimetres.

Fact	Multiply by
$1 m = 100 cm$	$100$
$1 m^{2} = 10000 cm^{2}$	$10 0^{2}$
$1 m^{3} = 1000000 cm^{3}$	$10 0^{3}$
$1 m^{2} = 1000000 mm^{2}$	$100 0^{2}$
$1 cm^{2} = 100 mm^{2}$	$1 0^{2}$
$1 cm^{3} = 1000 mm^{3}$	$1 0^{3}$
$1 km^{2} = 1000000 m^{2}$	$100 0^{2}$

Rule

Area: square the length factor. Volume: cube it. Then multiply to the smaller unit, divide to the bigger unit.

Textbooks print this as one formula. It works for any pair of units, even ones not in the table.

Formula

area factor = (length factor)^{2} volume factor = (length factor)^{3}

The length factor is how many of the small unit fit along one of the big unit, for example 10 for cm to mm. The area factor and volume factor are what you multiply or divide an area or a volume by.

Worked example

1
$7 cm^{2} \to mm^{2}$ — The question. A millimetre is smaller, so the answer will be a bigger number: multiply.
2
$1 cm = 10 mm$ — Find the length factor from the length facts: 10.
3
$10^{2} = 100$ — It is an area, so square the length factor. The area factor is 100.
4
$7 \times 100 = 700 mm^{2}$ — Multiply by the area factor, because the new unit is smaller.

Worked example

1
$2.5 m^{2} \to mm^{2}$ — A bigger jump, straight from square metres to square millimetres. Again multiply.
2
$1 m = 1000 mm$ — The length factor is 1000, because 1 m = 100 cm = 1000 mm.
3
$1000^{2} = 1000000$ — Square it for area. The area factor is one million.
4
$2.5 \times 1000000 = 2500000 mm^{2}$ — Multiply by one million: move each digit six places left.

Common mistake

Writing 7 cm² = 70 mm². That uses the length factor 10. For area it must be 10² = 100.

Area in Square Metres

The 2-mark question: work out an area from lengths in cm, then give it in square metres.

A table top is 45 cm by 80 cm. Find its area in square metres.

Worked example

1
$45 \times 80 = 3600 cm^{2}$ — Work out the area in the units you were given: length × width gives 3600 cm².
2
$3600 \div 10000$ — A square metre is bigger, so divide. The area factor is 100² = 10 000, not 100.
3
$= 0.36 m^{2}$ — Move each digit four places right. A table under half a square metre is sensible.

Key idea

Quicker: change the lengths first. $0.45 \times 0.8 = 0.36 m^{2}$ . Same answer.

Common mistake

Dividing by 100. That gives 36 square metres: a table as big as a classroom.

Which is larger? Area P = 42 000 cm². Area Q = 3.9 m². Exams say “show all your working”, so write the conversion down.

Worked example

1
$3.9 \times 10000 = 39000 cm^{2}$ — Put both areas in the same unit. Square cm is smaller, so multiply Q by the area factor 10 000.
2
$42000 cm^{2} > 39000 cm^{2}$ — Now compare the numbers. 42 000 is bigger, so Area P is larger.

Key idea

Tiny areas work the same way. $0.000 006 m^{2} \times 10000 = 0.06 cm^{2}$ : move the digits four places left, and count the zeros carefully.

Volume in Cubic Metres

The 1 or 2-mark question: change a volume between cubic centimetres and cubic metres. The factor is $10 0^{3} = 1000000$ .

Worked example

1
$3500000 cm^{3} \to m^{3}$ — The question. A cubic metre is bigger than a cubic centimetre, so divide.
2
$3500000 \div 1000000$ — It is a volume, so the factor is the length factor cubed: 100³ = 1 000 000.
3
$= 3.5 m^{3}$ — Move each digit six places right.

Worked example

1
$2.4 m^{3} \to cm^{3}$ — The question. A cubic centimetre is smaller, so multiply.
2
$2.4 \times 1000000$ — Use the same factor, 100³ = 1 000 000, because the units are the same pair.
3
$= 2400000 cm^{3}$ — Move each digit six places left, filling the gaps with zeros.

Common mistake

Multiplying by 100 or 10 000. A cube has three lengths, so use 100 three times.

Volume to Litres

The link between volume and liquid: one cubic centimetre holds one millilitre.

Volume	Holds
$1 cm^{3}$	1 ml
$1000 cm^{3}$	1 litre
$1 m^{3}$	1000 litres

A fish tank is 50 cm by 40 cm by 30 cm. How many litres does it hold?

Worked example

1
$50 \times 40 \times 30 = 60000 cm^{3}$ — Find the volume first: length × width × height gives 60 000 cm³.
2
$= 60000 ml$ — Each cubic centimetre holds 1 ml, so the number stays the same and only the unit changes.
3
$60000 \div 1000 = 60 litres$ — There are 1000 ml in a litre. A litre is bigger, so divide.

Common mistake

Thinking 1 cubic metre is 1 litre. A cubic metre is a big box: it holds 1000 litres.

Which Method? Read the Question

Check the unit you have and the unit the answer line wants.

You see	You do
$cm^{2} \leftrightarrow m^{2}$	× or ÷ 10 000
$mm^{2} \leftrightarrow cm^{2}$	× or ÷ 100
$m^{2} \leftrightarrow mm^{2}$	× or ÷ 1 000 000
$cm^{3} \leftrightarrow m^{3}$	× or ÷ 1 000 000
cm³ into litres	cm³ = ml, then ÷ 1000
Which area is larger?	change both to one unit, then compare
Lengths, masses, speeds	Unit Conversions

Key idea

Not sure? Change the lengths first, then work out the area or volume in the new unit.

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Real past-paper questions on area and volume conversions

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