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06 May 2015

Units of Volume

Some weeks ago I was involved in a long e-mail discussion over units of with American colleagues. Arguments went back and forth, but we could not agree on what Tf3 meant.

I finally came to the conclusion that Tf3 has a different meaning in the US. The unit prefixes used there, even if graphically similar, are not interpreted the same way as the prefixes of the International System of Units (ISU). These differences are likely unknown to most outside the US (like they were to me) and are possibly the source of many errors.

There is also a linguistic twist to this matter that is important to note.

Back in elementary school I learnt the meaning of a few small Greek words used to express measures of units of mass and length relevant in daily life:

  • m - mili - x 10-3
  • c - centi - x 10-2
  • d - deci - x 10-1
  • da - deca - x 10
  • h - hecta - x 102
  • k - kilo - x 103

The tricky part would then be areas and volumes, with each additional dimension the prefixes increase in magnitude. While a deca-meter is 10 times longer than a meter, a square with 1 deca-meter in side has 100 times the area of a square with 1 meter in side. It follows that a cube with a deca-meter in side contains 1 000 cubes of 1 meter in side.

To pass on to high-school I had to learn these relationships, which would be resumed in a table like this:
Length1 000100100.10.010.001
Area1 000 00010 0001000.010.00010.00001
Volume1 000 000 0001 000 0001 0000.0010.0000010.000000001

Thus when discussion broke up on the meaning of a figure expressed in Tf3 I interpreted it in light of the ISU: 1036 cubic feet. However my colleagues kept insisting that the Tera prefix equated only to 1012, be it lengths or volumes at stake. After a long exchange I finally concluded that unit prefixes have a different meaning in the United States. While I was interpreting this symbols as cubic-tera-feet, for the American colleagues it means tera-cubic-feet. So not only base units used there are different (e.g. feet instead of meters) but also prefixes. It follows that a unit such as 1 km3 in the US actually equates to only 1 dam3 in the rest of the world.

Therefore the conversion between American units and the USI is nothing but straightforward. One must be extremely careful interpreting or translating volume (or area) units originating in American literature. The table below offers a quick lookup.

1 ft30.028 m3
1 kft328.317 m3
1 Mft328.317 dam3
1 Gft328.317 hm3
1 Tft328.317 km3
1 m31 m3
1 km31 dam3
1 Mm31 hm3
1 Gm31 km3
1 Tm31 000 km3

There is an important linguistic aspect to this discussion that explains the different interpretations of units of volume. In Latin languages the characteristic of volume or area in a unit is expressed as an adjective, i.e. it becomes a suffix. For instance mêtre carré in French or metro cubo in Italian. Thus in these languages the symbol km3 will always be interpreted as kilómetro cúbico (in Castellan) or kilometru cub (Romanian), with prefix and suffix both unequivocally in place. Since the volumetric characteristic is always a suffix in these languages, the interpretation Americans give to the symbol km3 (kilo-cubic-meter) is actually not possible.

A literal translation of the American formulation kilo-cubic-meter, results in the same formulation as the USI interpretation of km3, e.g. kilo-metro-cúbico in Portuguese - the prefix and the suffix remain in place. Even if some sort of graphical distinction could be attempted, vocally the two are undistinguishable. In such cases a possible formulation to correctly translate these American expressions is to drop the ISU prefixes and use something like a thousand cubic meters (e.g. mil metros cúbicos in Portuguese).

Note however, that quantifiers such as billion, trillion or quadrillion also have different meanings in the US. E.g. billion-cubic-feet actually means thousand-million-cubic-feet, or 28.3 hm3 in the ISU. Confused? There is good reason to be so.

I suspect these differences from the American units system to the ISU are poorly known, providing fertile ground for all sorts of mistakes in translated literature. One of such mistakes seems to be relatively common: the usage of Gm3.

What is exactly a Gm3? At its aphelion Jupiter dists little over 0.8 Gm from the Sun. Therefore, a cube of 1 Gm3 would comfortably contain both celestial bodies and everything in between....