Units of area & volume

1 m is 100 cm, everybody knows that. So how many square centimetres are 1 m²? The most common answer is 100, and it is wrong. It is 10,000. This topic explains where the difference comes from, and afterwards you will convert areas and volumes as confidently as lengths.

Two sides, two conversions

Picture a square with a side of 1 m. Measured in centimetres each side is 100 cm long. The area is side times side, so $100 \cdot 100 = 10{,}000$ cm². The conversion factor 100 appears twice, because an area has two directions. That is exactly why units of area jump in steps of one hundred: 1 m² = 100 dm², 1 dm² = 100 cm², 1 cm² = 100 mm².

The three ladders

You only need to remember one number per kind of quantity. For lengths every step is a factor of 10, for areas 100, for volumes 1000. The order of the units always stays the same: mm, cm, dm, m. Going down the ladder the number grows, going up it shrinks, because smaller units make more pieces.

For plots of land and fields there are two in-between units that break the pattern. 1 are is a square with a side of 10 m, so 100 m². 1 hectare is a square with a side of 100 m, so 10,000 m². A football pitch is about 0.7 ha, and 100 ha make 1 km². For the hectare the picture helps: 1 ha is a square that a football pitch fits into pretty much exactly.

Volume and litres

For volume there is a second family of units that appears everywhere in daily life: litres. It is not a competitor but another name for the same quantities. 1 l = 1 dm³, 1 ml = 1 cm³, and 1000 l = 1 m³. That is why a cube with a 1 m edge holds exactly one cubic metre, so 1000 litres of water, and that weighs one tonne.