Volume, Capacity and Units

Volume answers carry a little raised three, and it is worth knowing what it stands for. One cubic centimetre is a genuine little cube, one centimetre along every edge — about the size of a sugar cube. You wrote the three because you multiplied three lengths together. One length gives centimetres, two gives square centimetres, three gives cubic centimetres, so that small number is really a headcount of how many lengths went into the answer. You will meet three of these units: cubic millimetres for tiny things, cubic centimetres for everyday objects, and cubic metres for large ones. Choose the one that gives a comfortable number, because a swimming pool measured in cubic centimetres runs to seven digits.

Volume and capacity describe the same space in two different ways: volume is how much room the solid takes up, capacity is how much liquid it can hold. Two conversions join them, and they are the heart of this chapter. One cubic centimetre holds exactly one millilitre. And a cube ten centimetres along each edge holds ten times ten times ten, which is a thousand cubic centimetres, and that is exactly one litre. So a thousand cubic centimetres is one litre. Whenever a question asks how many litres a container holds, find its volume in cubic centimetres and divide by a thousand. Going the other way, from litres to cubic centimetres, you multiply by a thousand, because cubic centimetres are the smaller unit.

Cubic metres are where people come unstuck. One metre is a hundred centimetres, so it is tempting to say one cubic metre is a hundred cubic centimetres. It is not. Picture a cube a metre along every edge and fill it with little centimetre cubes: a hundred along, a hundred back and a hundred up, which is a hundred times a hundred times a hundred — a million cubic centimetres. The factor gets cubed because there are three directions to fill. Divide that million by a thousand and you get a thousand litres, so one cubic metre holds a thousand litres. The safest habit of all is to convert your lengths to a single unit before you multiply, so this conversion never has to be done at the end.

Here is the whole routine on one question. A fish tank measures 50 centimetres long, 30 wide and 24 high, and you are asked how many litres it holds. First check the units: all three are already in centimetres, so nothing needs converting. Then find the volume: 50 times 30 times 24 is 36000 cubic centimetres. Then turn that into capacity by dividing by a thousand, giving 36 litres. Finally, spend ten seconds asking whether that sounds right. A bucket holds about 10 litres, so this tank is between three and four buckets of water, which is believable for a fish tank. If a question mixes centimetres and metres, that is deliberate — convert first, then multiply.

The numbers in these conversions are easy; the direction is what people get wrong. Here is a check that never fails. Moving to a smaller unit means you need more of them, so multiply — two litres becomes two thousand millilitres. Moving to a bigger unit means fewer, so divide — four and a half thousand millilitres becomes four point five litres. Run that check on every answer, because a smaller unit paired with a smaller number is impossible. The facts to learn are these: one cubic centimetre is one millilitre, a thousand cubic centimetres is a litre, a litre is a thousand millilitres, a cubic metre is a million cubic centimetres or a thousand litres, and a cubic centimetre is a thousand cubic millimetres.