C5.1; E5.1 - Metric units and conversions

Length and mass conversions

If $1\,\mathrm{m}=100\,\mathrm{cm}$, why is $1\,\mathrm{m}^2$ not $100\,\mathrm{cm}^2$? First separate the size of a unit from the number of units you need.

A desk has the same length whether you describe it as $1.2\,\mathrm{m}$ or $120\,\mathrm{cm}$. Centimetres are smaller units, so more of them fit along the desk. A conversion changes the number and the unit together; it does not change the actual length.

The standard length relationships are

For mass, $1\,\mathrm{kg}=1000\,\mathrm{g}$. Mass describes how much matter an object contains. A gram measures mass, so it cannot be converted into centimetres, which measure length.

To go to a smaller unit, multiply by the number of those smaller units in one larger unit. To reverse the conversion, divide by the same factor. A zero measurement remains zero; ordinary physical lengths and masses are non-negative.

Keep the same cable but reverse which unit is given. The factor is unchanged; the operation reverses.

The same reasoning applies to other metric units. For example, $0.64\,\mathrm{kg}=640\,\mathrm{g}$, $7.2\,\mathrm{km}=7200\,\mathrm{m}$, and $48\,\mathrm{mm}=4.8\,\mathrm{cm}$.

Use the unit size to check the direction of each conversion.

Square and cubic units

An area unit covers a surface; a volume unit fills space. Their conversion factors come from counting in two or three directions.

Imagine a square with side $1\,\mathrm{m}$. Each side is $100\,\mathrm{cm}$, so it contains $100$ rows of $100$ centimetre squares:

A cube of side $1\,\mathrm{m}$ has $100$ centimetre cubes along each edge and $100$ layers. Therefore

[DIAGRAM: asset_name: Units in two and three dimensions; asset_slug: square_cubic_conversion; caption: Count rows for area and layers as well for volume.; description: A square metre has both sides labelled 100 centimetres and the equation 100 times 100 equals 10000 square centimetres. Beside it a cube has three edges labelled 100 centimetres and the equation 100 times 100 times 100 equals 1000000 cubic centimetres. A note explains the grid divisions are grouped and not individual centimetre cells.]

Useful consequences are $1\,\mathrm{cm}^2=100\,\mathrm{mm}^2$, $1\,\mathrm{km}^2=1000000\,\mathrm{m}^2$, $1\,\mathrm{cm}^3=1000\,\mathrm{mm}^3$ and $1\,\mathrm{m}^3=1000000000\,\mathrm{mm}^3$. The exponent belongs to the whole unit, not to the measurement's numerical value.