10.01a - Converting units

10.01a - Converting units

Does changing metres to centimetres multiply an area by 100 too? Convert length, mass, area, volume, capacity, time and money without changing the quantity. Use the same reasoning when measurements contain letters, and see why square and cubic units need different factors.

Length and mass conversions

A square tile is 1 m1\,\mathrm{m} wide. Each edge is 100 cm100\,\mathrm{cm}, but does that mean its area is 100 cm2100\,\mathrm{cm}^2? Hold that question while we first make sense of changing the units of one length.

Unit conversion

A unit conversion changes how a quantity is expressed, not the quantity itself. A smaller unit needs a larger numerical count: the same ribbon contains more millimetres than centimetres.

The prefixes tell you useful relationships: “kilo” means a thousand, “centi” a hundredth, and “milli” a thousandth. For standard length and mass units, use these exact facts.

LengthMass
1 km=1000 m1\,\mathrm{km}=1000\,\mathrm{m}1 tonne=1000 kg1\text{ tonne}=1000\,\mathrm{kg}
1 m=100 cm=1000 mm1\,\mathrm{m}=100\,\mathrm{cm}=1000\,\mathrm{mm}1 kg=1000 g1\,\mathrm{kg}=1000\,\mathrm{g}
1 cm=10 mm1\,\mathrm{cm}=10\,\mathrm{mm}1 g=1000 mg1\,\mathrm{g}=1000\,\mathrm{mg}

Converting to smaller units means multiplying by the number of small units in one large unit. Reverse the conversion by dividing by the same factor. Do not choose multiplication merely because a question says “convert”.

Worked example 1: metres to centimetres

Question. Convert 2.75 m2.75\,\mathrm{m} to centimetres.

Plan. Count one hundred centimetres for every metre.

2.75×100=275 cmsmaller units2.75\times100=275\,\mathrm{cm} \qquad\text{smaller units}

Answer. 275 cm275\,\mathrm{cm}.

Check. 275÷100=2.75 m275\div100=2.75\,\mathrm{m}. Also, 2 m2\,\mathrm{m} and 0.75 m0.75\,\mathrm{m} contribute 200 cm200\,\mathrm{cm} and 75 cm75\,\mathrm{cm}.

Now reverse only the conversion direction. The number must decrease when the unit becomes larger.

Worked example 2: centimetres to metres

Question. Convert 480 cm480\,\mathrm{cm} to metres.

Plan. Group the centimetres into hundreds.

480÷100=4.8 mlarger units480\div100=4.8\,\mathrm{m} \qquad\text{larger units}

Answer. 4.8 m4.8\,\mathrm{m}.

Check. Four metres use 400 cm400\,\mathrm{cm}, leaving 80 cm=0.8 m80\,\mathrm{cm}=0.8\,\mathrm{m}.

Mass follows the same reasoning, with its own factor. For example, 0.064 kg=0.064×1000=64 g0.064\,\mathrm{kg}=0.064\times1000=64\,\mathrm{g}, and 850 mg=850÷1000=0.85 g850\,\mathrm{mg}=850\div1000=0.85\,\mathrm{g}. Each check reverses the multiplication or division. Kilograms to grams uses 1000, not the 100 used for metres to centimetres.

Watch out

4.8 m=0.048 cm4.8\,\mathrm{m}=0.048\,\mathrm{cm} is impossible: the same length needs more centimetres than metres. Use the unit relationship and a size check together.

Apply the larger-unit direction to a mass.

Area and volume units

An area counts unit squares, while a volume counts unit cubes. Changing the unit changes two dimensions for area and three for volume.

The opening tile is one metre along each side. There are 100 one-centimetre squares along each row and 100 rows. That is 100×100=10000100\times100=10000 squares, not 100.

[DIAGRAM: asset_name: Why area and volume factors are squared and cubed; asset_slug: squared-cubed-units; caption: A one-metre square contains 100 by 100 square centimetres. A one-metre cube contains 100 by 100 by 100 cubic centimetres.; description: Left panel shows a square with both sides labelled one metre equals one hundred centimetres. A schematic grid indicates 100 columns and 100 rows, giving 10000 square centimetres. Right panel shows a cube with three edges labelled one hundred centimetres, giving 1000000 cubic centimetres. The grid is schematic, with a note that not every unit square or cube is drawn.]
A one-metre square contains 100 by 100 square centimetres. A one-metre cube contains 100 by 100 by 100 cubic centimetres.

Square and cubic conversion factors

If one larger length unit equals kk smaller length units, then

1 larger square unit=k2 smaller square units1\text{ larger square unit}=k^2\text{ smaller square units} 1 larger cubic unit=k3 smaller cubic units1\text{ larger cubic unit}=k^3\text{ smaller cubic units}

Use the reciprocal factors in the reverse direction. These conversion rules are not on the 2027 formula sheet: understand them rather than relying on a sheet.

In particular, 1 m2=10000 cm21\,\mathrm{m}^2=10000\,\mathrm{cm}^2, 1 cm2=100 mm21\,\mathrm{cm}^2=100\,\mathrm{mm}^2, 1 m3=1000000 cm31\,\mathrm{m}^3=1000000\,\mathrm{cm}^3, and 1 cm3=1000 mm31\,\mathrm{cm}^3=1000\,\mathrm{mm}^3.

Worked example 3: square metres to square centimetres

Question. Convert 0.32 m20.32\,\mathrm{m}^2 to square centimetres.

Plan. Square the length factor of 100 before converting the area.

1002=10000two dimensions0.32×10000=3200 cm2count smaller squares\begin{aligned} 100^2&=10000 &&\text{two dimensions}\\ 0.32\times10000&=3200\,\mathrm{cm}^2 &&\text{count smaller squares} \end{aligned}

Answer. 3200 cm23200\,\mathrm{cm}^2.

Check. A rectangle measuring 1 m1\,\mathrm{m} by 0.32 m0.32\,\mathrm{m} has the same dimensions as 100 cm100\,\mathrm{cm} by 32 cm32\,\mathrm{cm}. Counting its rows and columns gives 100×32=3200100\times32=3200 square centimetres.

Change only the conversion direction. Divide by the squared factor.

Worked example 4: square centimetres to square metres

Question. Convert 7500 cm27500\,\mathrm{cm}^2 to square metres.

Plan. Group the square centimetres into groups of 10000.

7500÷10000=0.75 m27500\div10000=0.75\,\mathrm{m}^2

Answer. 0.75 m20.75\,\mathrm{m}^2.

Check. 75007500 is three quarters of the 1000010000 square centimetres in one square metre.

Capacity is the volume a container can hold. Its common units link exactly to cubic units:

1 litre=1000 ml=1000 cm31\,\mathrm{litre}=1000\,\mathrm{ml}=1000\,\mathrm{cm}^3 1 ml=1 cm3,1 m3=1000 litres1\,\mathrm{ml}=1\,\mathrm{cm}^3,\qquad 1\,\mathrm{m}^3=1000\,\mathrm{litres}

A cubic centimetre is not a square centimetre: one measures space, the other surface.

Worked example 5: cubic metres to litres

Question. Convert 0.018 m30.018\,\mathrm{m}^3 to cubic centimetres and then litres.

Plan. Cube the length factor, then use 1000 cubic centimetres per litre.

0.018×1003=18000 cm3three dimensions18000÷1000=18 litrescapacity link\begin{aligned} 0.018\times100^3&=18000\,\mathrm{cm}^3 &&\text{three dimensions}\\ 18000\div1000&=18\,\mathrm{litres} &&\text{capacity link} \end{aligned}

Answer. 18000 cm3=18 litres18000\,\mathrm{cm}^3=18\,\mathrm{litres}.

Check. 0.018×1000=180.018\times1000=18 litres directly, since a cubic metre holds 1000 litres.

Reverse only the cubic-unit direction in the next example.

Worked example 6: litres to cubic metres

Question. Convert 24 litres24\,\mathrm{litres} to cubic centimetres and then cubic metres.

Plan. Turn litres into cubic centimetres, then divide by the cubic factor.

24×1000=24000 cm3capacity link24000÷1003=0.024 m3larger cubes\begin{aligned} 24\times1000&=24000\,\mathrm{cm}^3 &&\text{capacity link}\\ 24000\div100^3&=0.024\,\mathrm{m}^3 &&\text{larger cubes} \end{aligned}

Answer. 24000 cm3=0.024 m324000\,\mathrm{cm}^3=0.024\,\mathrm{m}^3.

Check. 0.024×1000=240.024\times1000=24 litres, recovering the starting capacity.

The opening tile therefore has area 10000 cm210000\,\mathrm{cm}^2. A length factor of 100 becomes an area factor of 10000 and a volume factor of 1000000.

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