Units of Area and Converting Between Them

Before converting anything, be clear about why area units carry that little raised 2. When you find the area of a rectangle 4 centimetres by 3, you multiply the numbers to get 12, but you also multiply the units: centimetres times centimetres gives square centimetres. Those 12 squares in the picture really are one centimetre by one centimetre each, so the answer counts squares rather than measuring a distance. That is why reading the unit tells you what kind of answer you have — plain units are lengths, squared units are areas. It also explains everything in this lesson: because two lengths were multiplied together, any conversion factor has to be used twice.

Here is a square centimetre drawn large. It is 1 centimetre across, which is 10 millimetres, and 1 centimetre down, which is another 10 millimetres. So it holds 10 rows of 10 tiny millimetre squares — 100 of them altogether. That means one square centimetre equals 100 square millimetres. Almost everyone gets this wrong the first time, because one centimetre is 10 millimetres so they write that one square centimetre is 10 square millimetres. Look at the picture: ten little squares would be a single row along the top, nowhere near the whole thing. The square grew ten times wider and ten times taller, two directions, so the ten is used twice. And five square centimetres is 500 square millimetres.

Now the full picture. Take any length conversion you already know and square the factor to get the area one. One centimetre is 10 millimetres, so one square centimetre is 10 times 10, which is 100 square millimetres. One metre is 100 centimetres, so one square metre is 100 times 100, which is 10000 square centimetres. One kilometre is 1000 metres, so one square kilometre is a million square metres. Then the direction is the same as always: going to a smaller unit you need more of them, so multiply — 3 square metres is 30000 square centimetres. Going to a bigger unit you need fewer, so divide — 25000 square centimetres is 2.5 square metres.

Land is too big for square metres and too small for square kilometres, so there is a unit in between: the hectare. One hectare is a square 100 metres by 100 metres, which is 10000 square metres — roughly two football fields. Because a square kilometre is 1000 metres each way, it holds ten rows of ten hectares, so one square kilometre is 100 hectares. You can check that: 100 hectares times 10000 is a million square metres, which matches. To go from square metres to hectares you divide by 10000, and to go back you multiply by 10000. So a sports field 120 metres by 75 covers 9000 square metres, which is 0.9 of a hectare — just under one.

Finally, how this shows up in real questions. A rectangle is 2 metres by 50 centimetres, and you cannot multiply those as they stand — 2 times 50 gives 100 of a unit that does not exist. Convert first. Working in centimetres, 2 metres is 200, so the area is 200 times 50, which is 10000 square centimetres. Working in metres, 50 centimetres is 0.5, so the area is 2 times 0.5, which is 1 square metre. Both describe the same rectangle, and since 10000 square centimetres is exactly 1 square metre they agree — a free check on your factor. Which unit to choose? The one the question asks for, or the one that gives a comfortable number.