Area and Volume Unit Conversions

In the exam you may be asked to change an area or a volume into a different unit, such as square metres into square centimetres, for 1 mark. Examiners say these are some of the worst-answered questions on the Foundation papers, because many students use 100 or 1000 instead of 10\,000

Longer questions, for 3 to 5 marks, are about a tank or a container. You work out its volume, change it into litres, then use it, for example with a rate in litres per minute.

The question sometimes gives a fact such as 1 litre =1000\text{ cm}^3, but not always, so learn the facts on this page. Length conversions, such as metres to centimetres, are on the Metric Units page.

Area is measured in square units. One square centimetre, 1\text{ cm}^2, is the area of a square 1 cm long and 1 cm wide.

A square metre is a square 1 m long and 1 m wide. Each side is 100 cm, so it holds 100 rows of 100 small squares, each 1 cm by 1 cm.

So a square metre is not 100\text{ cm}^2. It is

1\text{ m}^2=100\times100=10\,000\text{ cm}^2

In the same way, 1 cm is 10 mm, so

1\text{ cm}^2=10\times10=100\text{ mm}^2

Changing to a smaller unit, you need more of them, so the number gets bigger. Multiply: 0.35\text{ m}^2 is 0.35\times10\,000=3500\text{ cm}^2

Changing to a bigger unit, you need fewer of them, so the number gets smaller. Divide: 2500\text{ mm}^2 is 2500\div100=25\text{ cm}^2

Multiplying by 10\,000 moves every digit four places to the left. Dividing by 10\,000 moves every digit four places to the right, so 62\,000\div10\,000=6.2

If you forget the number, sketch the square: multiply the length conversion by itself. Metres to centimetres is 100 for a length, so it is 100\times100 for an area.

Volume is measured in cubic units. A cube with edges of 1 cm is 10 mm long, 10 mm wide and 10 mm high, so

1\text{ cm}^3=10\times10\times10=1000\text{ mm}^3

A cubic metre has edges of 100 cm, so

1\text{ m}^3=100\times100\times100=1\,000\,000\text{ cm}^3

To change \text{m}^3 into \text{cm}^3, multiply by 1\,000\,000. To change \text{cm}^3 into \text{m}^3, divide by 1\,000\,000. Dividing by 1\,000\,000 is the same as dividing by 1000 and then by 1000 again.

Capacity is how much a container holds, often in litres or millilitres. These facts link capacity to volume:

1\text{ litre}=1000\text{ cm}^3

1\text{ ml}=1\text{ cm}^3

1\text{ m}^3=1000\text{ litres}

A cube 10 cm by 10 cm by 10 cm has a volume of 1000\text{ cm}^3, and it holds exactly 1 litre. So to change \text{cm}^3 into litres, divide by 1000.

A cubic metre is 1\,000\,000\text{ cm}^3, and 1\,000\,000\div1000=1000, so 1\text{ m}^3 holds 1000 litres. To change \text{m}^3 into litres, multiply by 1000.

Millilitres and cubic centimetres are the same size, so 250 ml is 250\text{ cm}^3

To work out an area or a volume, every length must be in the same unit. A board 3 m long and 20 cm wide does not have an area of 3\times20=60 of anything.

The safest way is to change the lengths into the unit the answer needs, then multiply. A length uses the easy numbers 10, 100 and 1000.

You can also work out the area in one unit and change it at the end. Then use 10\,000 for square metres to square centimetres, never 100.

• Using 100 to change \text{m}^2 into \text{cm}^2. A square metre is 100\times100=10\,000\text{ cm}^2
• Using 10 or 100 to change \text{cm}^3 into \text{mm}^3. A cubic centimetre is 10\times10\times10=1000\text{ mm}^3
• Moving the digits the wrong number of places. 5000\text{ cm}^3 is 5000\div1\,000\,000=0.005\text{ m}^3, so count the places carefully.
• Multiplying lengths in different units. Change every length into the same unit before you work out an area or a volume.
• Using the wrong number for litres. 1 litre =1000\text{ cm}^3 and 1\text{ m}^3=1000 litres.
• Multiplying by a rate instead of dividing. Time is the amount of water divided by the litres per minute.