M4.5.5 Chapter 5

Changing units of area and volume

How many square centimetres are in one square metre?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A 100 square centimetres.
You picked A. Read on. At the end, the chapter comes back to your guess.
B 1 000 square centimetres.
You picked B. Read on. At the end, the chapter comes back to your guess.
C 10 000 square centimetres.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Change between cm² and m², and between cm³ and m³
Change between cm³, litres and m³

You know that 1 m = 100 cm. It is tempting to think that 1 m² = 100 cm² as well. It is not. This chapter shows why, and gives one rule that works for every unit.

A square metre is the area of a square 1 m long and 1 m wide. We can cut that square into smaller squares, 10 cm by 10 cm. The picture shows them.

One square metre, cut into 100 squares of 10 cm by 10 cm.

Each small square is 10 cm by 10 cm, so its area is 10 × 10 = 100 cm². There are 10 rows of 10, which is 100 small squares. So the whole area is 100 × 100 = 10 000 cm².

The same idea works for any unit. The square has 100 cm along each edge, and the area is 100 × 100. So 1 m² = 10 000 cm². Likewise 1 cm = 10 mm, so 1 cm² = 10 × 10 = 100 mm².

The area 3.5 m² is three whole square metres and half of another.

Worked example

Change 3.5 m² into cm².

One square metre is 10 000 cm².
There are 3.5 square metres, so multiply: 3.5 × 10 000.
3.5 × 10 000 = 35 000 cm².

Answer: 35 000 cm²

To change cm² into m², go the other way and divide by 10 000. A small unit gives a big number, and a big unit gives a small number. So 20 000 cm² is 2 m².

Now volume. A cube 1 m on each edge has volume 1 m³. It is also 100 cm by 100 cm by 100 cm. The picture cuts it into small cubes, 10 cm on each edge.

One cubic metre. It holds 10 × 10 × 10 = 1 000 small cubes.

A cube with 100 cm along each edge has volume 100 × 100 × 100 = 1 000 000 cm³. So 1 m³ = 1 000 000 cm³. A million small cubes, each 1 cm on an edge, fill a cubic metre.

When lengths change by a factor, areas change by that factor squared and volumes by that factor cubed.

The table gives the rule. Change the unit of length by a factor k. The unit of area changes by k × k. The unit of volume changes by k × k × k. For m to cm, k = 100.

A litre is a unit for liquids. One litre is the volume of a cube 10 cm on each edge. That is 10 × 10 × 10 = 1 000 cm³. One millilitre (ml) is one thousandth of a litre, so 1 ml = 1 cm³.

One litre is a cube 10 cm on each edge.

The cubic metre picture held 1 000 small cubes of 10 cm. Each is one litre. So 1 m³ = 1 000 litres. It also agrees with 1 000 000 cm³ ÷ 1 000 cm³ = 1 000.

Worked example

Lucía has a fish tank that is 80 cm long, 40 cm wide and 50 cm high. How many litres does it hold when it is full?

The volume is length × width × height = 80 × 40 × 50.
80 × 40 × 50 = 160 000 cm³.
One litre is 1 000 cm³, so divide by 1 000.
160 000 ÷ 1 000 = 160 litres.

Answer: 160 litres

The fish tank, full to the top.

A water store 1 m by 1 m by 1 m, filled to 75 cm. It holds 0.75 m³.

Worked example

A water store holds 0.75 m³. How many cm³ is that, and how many litres?

One cubic metre is 1 000 000 cm³.
So 0.75 m³ is 0.75 × 1 000 000 = 750 000 cm³.
One litre is 1 000 cm³. So 750 000 cm³ is 750 000 ÷ 1 000 = 750 litres.
Check with 1 m³ = 1 000 litres: 0.75 × 1 000 = 750 litres. The two ways agree.

Answer: 750 000 cm³ and 750 litres

Try it

A hundred small squares

You need: squared paper with 1 cm squares, a pencil and a ruler.

Draw a square 10 cm by 10 cm on the paper.
Count the squares inside it. Count the rows, then the columns.
Work out 10 × 10 and compare with your count.
Imagine 100 rows of 100 of these small squares. That is one square metre.

What you will see: The square holds 100 squares of 1 cm², so its area is 100 cm². A square 10 times as long on each side, 100 cm, would hold 10 × 10 times as many, which is 10 000.

If you cannot try it, this is what happens.

A wrong idea many people have

Since 1 m = 100 cm, it follows that 1 m² = 100 cm².

Why it feels right: We change metres to centimetres by multiplying by 100, and the little 2 only seems to say that it is an area.

The test that breaks it: Draw a square 1 m by 1 m. In centimetres it is 100 cm by 100 cm, and its area is 100 × 100 = 10 000 cm². It is not 100 cm². A 10 cm by 10 cm square has area 100 cm², which is a tiny part of a square metre.

What is true: Areas change by the square of the length factor, and volumes by the cube. So 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³.

A wrong idea many people have

A litre is 100 cm³, because a litre box is 10 cm long.

Why it feels right: The edge of the litre cube is 10 cm, and 10 × 10 = 100 is a number that is quick to remember.

The test that breaks it: A litre cube has three edges of 10 cm. Volume is length × width × height, so 10 × 10 × 10 = 1 000 cm³. A square of 10 × 10 is only a flat face of the cube.

What is true: One litre is 1 000 cm³, so 1 ml is 1 cm³. And 1 000 litres make 1 m³.

Key words

litre: a unit for liquids: the volume of a cube 10 cm on each edge, which is 1 000 cm³.
millilitre: one thousandth of a litre. 1 ml is the same as 1 cm³.

Back to your guess

The answer is C: 10 000 square centimetres.

A square metre is 100 cm long and 100 cm wide, so it holds 100 × 100 = 10 000 square centimetres. The number is not 100, because 100 is the number of centimetres along one edge.

Keep it

Copy this into your exercise book.

1 m² = 10 000 cm². 1 m³ = 1 000 000 cm³. 1 litre = 1 000 cm³. 1 ml = 1 cm³. 1 m³ = 1 000 litres. Small unit to big unit: divide.

Draw: A square metre cut into 100 squares of 10 cm, with one square shaded and labelled 100 cm².