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Converting area and volume units

Measurement · level 5, lesson 5 of 6

A question to start

There are 100 centimetres in a metre. So how many little squares, each 1 cm by 1 cm, does it take to cover a square that is 1 m by 1 m?

The tempting answer is 100. The real answer is 10 000. And how many centimetre cubes fill a box 1 m along each edge? Not 100, not 10 000, but one million.

If you get these wrong, a floor plan can be out by a factor of 100, and a water tank by a factor of 10 000. This lesson shows why the numbers jump so fast, and how to convert areas and volumes without ever getting it wrong.

The idea

Lengths: one factor

From level 3 you know the metric length family:

mm cm m km
1 cm is 10 mm
1 m is 1000 mm 100 cm
1 km is 1000 m

To change metres to centimetres you multiply by 100, because each metre is 100 centimetres laid end to end.

Areas: the factor twice

Now draw a square 1 m by 1 m and divide each side into ten parts of 10 cm. The square is cut into a 10 by 10 grid of smaller squares, each 10 cm by 10 cm.

each small square: 10 cm × 10 cm = 100 cm² 10 rows of 10 squares: 100 squares 100 × 100 cm² = 10 000 cm² 1 m = 100 cm

There are 100 small squares, and each one holds \(10 \times 10 = 100\) square centimetres. So the big square holds \(100 \times 100 = 10\,000\) cm². Another way to see it: a 1 m square is 100 cm long and 100 cm wide, so it is 100 rows of 100 squares.

\[1 \text{ m}^2 = 100 \text{ cm} \times 100 \text{ cm} = 10\,000 \text{ cm}^2\]

The reason is that area is a length times a length. When each length is multiplied by 100, the area is multiplied by \(100 \times 100\). The little 2 in "m²" is a reminder: square the length factor.

Volumes: the factor three times

A cube 1 m along each edge is 100 cm long, 100 cm wide and 100 cm tall. Filling it layer by layer (level 4): each layer is \(100 \times 100 = 10\,000\) cubes, and there are 100 layers.

\[1 \text{ m}^3 = 100 \times 100 \times 100 \text{ cm}^3 = 1\,000\,000 \text{ cm}^3\]

Volume is length times length times length, so the length factor is used three times: cube the factor. The little 3 in "m³" is the reminder.

The general rule

If one unit of length is \(k\) of a smaller unit, then

  • one unit of area is \(k^2\) of the smaller square unit;
  • one unit of volume is \(k^3\) of the smaller cubic unit.
Length Area Volume
1 cm = 10 mm 1 cm² = 100 mm² 1 cm³ = 1000 mm³
1 m = 100 cm 1 m² = 10 000 cm² 1 m³ = 1 000 000 cm³
1 m = 1000 mm 1 m² = 1 000 000 mm² 1 m³ = 1 000 000 000 mm³
1 km = 1000 m 1 km² = 1 000 000 m² 1 km³ = 1 000 000 000 m³

You never need to memorise this table. Write the length fact, then square it or cube it.

As in level 3, converting to a smaller unit gives a bigger number (multiply); converting to a bigger unit gives a smaller number (divide). Ask yourself: "are there going to be more of these units, or fewer?"

Hectares: a unit for land

Square metres are too small for farms and parks, and square kilometres are too big for a field. So land is often measured in hectares (ha):

\[1 \text{ ha} = 100 \text{ m} \times 100 \text{ m} = 10\,000 \text{ m}^2\]

A hectare is a square 100 m on each side, about the size of a large city block. A full-size soccer field (105 m by 68 m) is about 0.7 ha. Because 1 km is 10 lengths of 100 m, a square kilometre holds \(10 \times 10 = 100\) hectares:

\[1 \text{ km}^2 = 100 \text{ ha}\]

(The word comes from "hecto", meaning 100, and "are", an old unit of 100 m². So a hectare is 100 ares.)

Litres join the family

From level 4: a litre is the volume of a cube 10 cm on each edge, \(10 \times 10 \times 10 = 1000\) cm³, and a millilitre is 1 cm³. Since 1 m is 10 lengths of 10 cm, a cubic metre holds \(10^3 = 1000\) of those litre cubes:

\[1 \text{ mL} = 1 \text{ cm}^3 \qquad 1 \text{ L} = 1000 \text{ cm}^3 \qquad 1 \text{ m}^3 = 1000 \text{ L}\]

The safest method: convert the lengths first

When a problem gives lengths in mixed units, change all the lengths to one unit before multiplying. Then there is nothing to square or cube afterwards. If you must convert a finished area or volume, use the squared or cubed factor.

Worked examples

1. Floor area. A floor has an area of 3.5 m². How many cm² is that?

\(3.5 \times 10\,000 = 35\,000\) cm². (More units, because cm² are smaller: multiply.)

2. A big tank. A tank holds 2 500 000 cm³. How many cubic metres and how many litres?

  1. \(2\,500\,000 \div 1\,000\,000 = 2.5\) m³.
  2. \(2.5 \times 1000 = 2500\) L. (Check: \(2\,500\,000 \div 1000 = 2500\) L directly. ✓)

3. A field. A field is 250 m by 180 m. Find its area in hectares.

  1. \(250 \times 180 = 45\,000\) m².
  2. \(45\,000 \div 10\,000 = 4.5\) ha.

4. Mixed units, two ways. A box is 2 m long, 50 cm wide and 30 cm tall. Find its volume in m³ and in litres.

  • Lengths first, in metres: \(2 \times 0.5 \times 0.3 = 0.3\) m³, which is \(0.3 \times 1000 = 300\) L.
  • Or in centimetres: \(200 \times 50 \times 30 = 300\,000\) cm³, and \(300\,000 \div 1\,000\,000 = 0.3\) m³. ✓

5. A phone screen. A screen is 70 mm by 150 mm. Its area is \(70 \times 150 = 10\,500\) mm². Since 1 cm² = 100 mm², that is \(10\,500 \div 100 = 105\) cm².

6. Mars in hectares. The surface of Mars is about 144.8 million km², which is \(1.448 \times 10^8\) km² in scientific notation. Each km² is 100 ha, so Mars's surface is about \(1.448 \times 10^{10}\) hectares: roughly 14 billion hectares.

Try it

  1. Change 4 m² to cm².
  2. Change 6000 cm² to m².
  3. Change 0.75 m³ to cm³, and then to litres.
  4. Change 12 cm³ to mm³.
  5. A farm covers 3.2 km². How many hectares is that? How many m²?
  6. A house lot is 40 m by 25 m. What fraction of a hectare is it?
  7. An aquarium is 80 cm long, 35 cm wide and 40 cm deep. Find its volume in cm³, m³ and litres.
  8. A model of a house is built at a scale of 1 : 50 (every real length is 50 times the model's). The model's floor is 20 cm by 16 cm. What is the real floor area, in m²? Do it two ways: by converting lengths first, and by using the squared factor.
Answers
  1. \(4 \times 10\,000 = 40\,000\) cm².
  2. \(6000 \div 10\,000 = 0.6\) m².
  3. \(0.75 \times 1\,000\,000 = 750\,000\) cm³; and \(750\,000 \div 1000 = 750\) L (or \(0.75 \times 1000 = 750\) L).
  4. 1 cm³ = \(10^3 = 1000\) mm³, so \(12 \times 1000 = 12\,000\) mm³.
  5. \(3.2 \times 100 = 320\) ha. Each ha is 10 000 m², so \(320 \times 10\,000 = 3\,200\,000\) m².
  6. \(40 \times 25 = 1000\) m², and \(1000 \div 10\,000 = 0.1\): one tenth of a hectare.
  7. \(80 \times 35 \times 40 = 112\,000\) cm³ \(= 0.112\) m³ \(= 112\) L.
  8. Lengths first: real floor \(20 \times 50 = 1000\) cm = 10 m by \(16 \times 50 = 800\) cm = 8 m, area 80 m². Squared factor: model area \(20 \times 16 = 320\) cm², times \(50^2 = 2500\) gives \(800\,000\) cm², and \(800\,000 \div 10\,000 = 80\) m². ✓

Common mistake

Using the length factor for an area or a volume. "1 m = 100 cm, so 3 m² = 300 cm²" is wrong. It should be \(3 \times 10\,000 = 30\,000\) cm². Sketch a square and grid it if you are not sure.

Multiplying mixed units. "2 m × 50 cm × 30 cm = 3000" mixes metres and centimetres and gives a meaningless number. Make every length the same unit first.

Where it leads