Year 8MeasurementMeasuringYear 8 Maths

Volume, capacity and converting units

What this lesson covers

Liquids can be measured by capacity and by volume; there are standard conversions between measurements, in particular 1 mL = 1 cm³, 1L = 1000 cm³, and 1 m³ = 1000 L.

Key idea

Two words that sound like one thing

Volume is how much space something takes up.
Capacity is how much a container can hold.
Related, and measured in different units. Usually.

Word	Means	Measured in
Volume	space a thing takes up	cm³, m³
Capacity	what a container holds	mL, L

Key idea

And the bridge between them

One millilitre is exactly one cubic centimetre.
Not roughly. Exactly, by definition.
So capacity and volume are the same quantity, relabelled.

Word	Means	Measured in
Volume	space a thing takes up	cm³, m³
Capacity	what a container holds	mL, L
Bridge	1 mL is exactly 1 cm³

Key idea

Start with lengths, which you know

A centimetre is 10 millimetres.
A metre is 100 centimetres.
A kilometre is 1000 metres. Nothing new yet.

Unit		Equals
1 cm		10 mm
1 m		100 cm
1 km		1000 m

Key idea

So is a square centimetre 10 square millimetres?

It is the obvious guess, and it is wrong.
Draw the square centimetre and cut it into millimetre squares.
Count them. There are a hundred.

Key idea

Because a square has two directions

It is 10 millimetres across and 10 millimetres down.
10 times 10 is 100.
The factor got used twice, because area has two dimensions.

Question	Working	Answer
1 cm is 10 mm	so is 1 cm² = 10 mm²?	no
Count the grid	10 rows of 10	100
1 cm²		100 mm²

Key idea

One rule for everything that follows

For lengths, use the factor once.
For areas, twice. For volumes, three times.
The little 2 or 3 in the unit tells you how many.

Measuring	Then	Factor
Length	use the factor once	× 10
Area	use it twice	× 10 × 10 = 100
Volume	use it three times	× 10 × 10 × 10 = 1000

Key idea

Which gives you every area conversion

cm² to mm²: 10 squared, so times 100.
m² to cm²: 100 squared, so times 10 000.
km² to m²: 1000 squared, so times a million.

Going from	Because	Multiply by
cm² to mm²	10²	× 100
m² to cm²	100²	× 10 000
km² to m²	1000²	× 1 000 000

Key idea

And every volume one

cm³ to mm³: 10 cubed, so times 1000.
m³ to cm³: 100 cubed, which is a million.
Nothing memorised. All of it rebuilt from 10, 100 and 1000.

Going from	Because	Multiply by
cm³ to mm³	10³	× 1000
m³ to cm³	100³	× 1 000 000

Key idea

Going back, you divide by the same number

mm² to cm² is dividing by 100, not by 10.
The factor does not change direction. Only the operation does.
300 mm² is 3 cm². Not 30.

Conversion	Do	Example
cm² to mm²	× 100	3 cm² = 300 mm²
mm² to cm²	÷ 100	300 mm² = 3 cm²

Key idea

Now build a cube

Start with a square 10 centimetres each way.
That face is 100 square centimetres.
Now stack ten of those layers on top of each other.

Worked example

That cube holds exactly one litre

10 × 10 × 10 is 1000 cubic centimetres.
And a cubic centimetre is a millilitre, so 1000 millilitres.
Which is one litre. Check it against a milk bottle.

Step	Working	Gives
A cube 10 cm each way	10 × 10 × 10	1000 cm³
And 1 cm³ is 1 mL	so that is	1000 mL
Which is		1 litre

Worked example

And a cubic metre holds a thousand

A metre is 100 centimetres, so 100 cubed is a million cm³.
A million cubic centimetres is a million millilitres.
Divide by 1000 and that is 1000 litres.

Step	Working	Gives
1 m³	in cubic centimetres	1 000 000 cm³
That is	in millilitres	1 000 000 mL
So 1 m³		1000 L

Worked example

Three conversions, start to finish

5 m² to cm²: the factor is 100 squared, so 50 000.
2.5 litres to cm³: times 1000, so 2500.
3 m³ to litres: times 1000, so 3000.

Convert	Working	Answer
5 m² to cm²	5 × 10 000	50 000 cm²
2.5 L to cm³	2.5 × 1000	2500 cm³
3 m³ to L	3 × 1000	3000 L

Worked example

One going the other way

250 cubic centimetres. That is 250 millilitres.
To get litres, divide by 1000.
0.25 litres. A small glass of something.

Step	Working	Gives
250 cm³	same as	250 mL
250 mL	÷ 1000	0.25 L

Key idea

Picking a unit that fits

A swimming pool in millilitres is a silly number.
A dose of medicine in litres is worse.
Choose the unit that gives a number you can say out loud.

Measuring	Not	Use
A swimming pool	not litres	m³
A dose of medicine	not litres	mL
A farm	not m²	hectares or km²
A pencil lead	not cm	mm

Key idea

And having something to compare against

A litre is that ten centimetre cube. A milk bottle.
A square metre is about a small table top.
If your answer is wildly off those, check the factor.

Unit	Roughly
1 litre	a milk bottle
1 m²	a small table top
1 m³	a large fridge

Your turn

Your turn: four questions

Pause here and try all four.
For each one, ask whether it is area or volume first.
Question 4 goes backwards, so you will be dividing.

	Question
1	Convert 4 m² into cm².
2	Convert 1.5 L into cm³.
3	Convert 2 m³ into litres.
4	Convert 300 mm² into cm².

Show answer

1. 40 000 cm². The factor is 100 squared, which is 10 000. 2. 1500 cm³. A litre is 1000 cm³. 3. 2000 L. A cubic metre is 1000 litres. 4. 3 cm². Going the other way you divide, and the factor is still 100, not 10. The commonest wrong answer is 30.

Worked example

The answers

4 m² is 40 000 cm². And 1.5 litres is 1500 cm³.
2 m³ is 2000 litres.
300 mm² is 3 cm². Dividing by 100, not by 10.

	Answer
1	40 000 cm²
2	1500 cm³
3	2000 L
4	3 cm²

Summary

What to take away

Square the length factor for area, cube it for volume.
A millilitre is exactly a cubic centimetre.
And a ten centimetre cube holds exactly one litre.

	Remember
Area	square the length factor
Volume	cube it
1 mL	is exactly 1 cm³
A 10 cm cube	holds exactly 1 litre
1 m³	is 1000 litres

Is your child on track?

Take the free placement test