Area and Volume Unit Conversions
Why area squares the factor
Lengths convert cleanly: a metre is a hundred centimetres,
1\ \text{m} = 100\ \text{cm}. It is tempting to think a square
metre is therefore a hundred square centimetres — but that is wrong.
A 1\ \text{m} square is
100\ \text{cm} along each side, so it holds a
100 \times 100 grid of one-centimetre squares:
1\ \text{m}^2 = 100 \times 100 = 10\,000\ \text{cm}^2
Both sides shrink by the same factor, so the count of unit squares shrinks by that factor
twice. To convert an area you
square the length conversion factor.
Volume cubes it
A solid has three directions, so the same reasoning applies a third time. A
1\ \text{m} cube is 100\ \text{cm}
wide, deep and tall:
1\ \text{m}^3 = 100 \times 100 \times 100 = 1\,000\,000\ \text{cm}^3
So a length factor of k becomes k^2
for area and k^3 for volume — square it for flat shapes, cube it
for solids.
If a length conversion multiplies by k, then:
- an area multiplies by k^2;
- a volume multiplies by k^3.
Common cases:
- 1\ \text{cm}^2 = 100\ \text{mm}^2 and
1\ \text{m}^2 = 10\,000\ \text{cm}^2;
- 1\ \text{cm}^3 = 1000\ \text{mm}^3 and
1\ \text{m}^3 = 1\,000\,000\ \text{cm}^3.
Seeing the square metre
Each side of this square is 100\ \text{cm}. The gridlines split it
into rows and columns of one-centimetre squares — count them and there are
100 \times 100 = 10\,000.