Gas Mileage, with a Side of British Units

In the United States we measure how much gasoline an automobile uses in units of “miles per gallon”, often referred to as the car’s “fuel economy”. Elsewhere in the world it is measured in “liters (litres) per 100 kilometers”.

Since both gallons and liters are volume measurements when we do a dimensional analysis of these quantities, we get for the United States L/(L^3) or L^{-2}, and for the rest of the world L^3/L or L^2. In both cases it comes out as an area measurement, inverted in the case of the United States.

For 25mpg in the US (which is 9.4 liters per 100 kilometers), we can calculate the area knowing that a US gallon is defined to be 231 cubic inches, and since a mile is 5,280 feet, or 63,360 inches, the area (un-inverted) in square inches is: 

    \[\frac{231}{25\times 63360} = 0.000145833\]

 or a square 0.0121 inches on a side or a circle that is 0.01363 inches in diameter. In metric units, given that a liter is 1,000 cubic centimeters, or 1,000,000 cubic millimeters, and a kilometer is 1,000,000 millimeters, the area is 0.094 square millimeters, which is a square 0.3066 millimeters on a side, or a circle 0.3460 millimeters in diameter.

So that’s the area. But what does it mean physically?  It is the volume of gasoline that it takes to drive 25 miles, or in the other units 100 kilometers, stretched out to that length. If we form it into a very long cylinder then the area is the cross section of the cylinder. So a 25mpg automobile has the contents of its gas tank stretched out over the whole length of its journey, into a cylinder of gasoline with diameter 0.346 millimeters, and the car is precisely eating that cylinder as it drives along!

Of course, having grown up in Australia back when we used Imperial British units for everything, I have always preferred expressing gas mileage in acres^{\big 1}. And 25mpg turns out to be 2.325 \times 10^{-11} acres.

A Boeing 747 burns about 5 gallons of fuel per mile, or 12 liters per kilometer, so it is eating up a cylinder with a cross sectional area of 12 square millimeters, which is a cylinder with a 3.9 millimeter diameter, roughly 100 times more than an automobile.

The first stage (S1-C) of the Saturn V moon rockets burned out at about 61 kilometers up, having consumed 770,000 liters of RP-1 kerosene. That means it consumed a cylinder of fuel with 12,623 square millimeters, i.e., a diameter of 126.8 millimeters, or just about exactly five inches.  Now that is a gas guzzler!

^{\big 1}What is an acre? It is derived from the amount of land tillable by a yoke of oxen in one day–and a long strip of land is more efficient to till than a more boxy area as you have to change direction less often. So an acre is defined as one “chain” wide, by 10 chains long. And a chain?  It is a 100 links, or exactly 22 yards (and also exactly four “rods” long, each of which is sixteen and a half feet long). So the standard tillable plot was 22 yards wide, and 220 yards long, which happens to be one eighth of a mile long, otherwise known to horse racing enthusiasts as a furlong, or “furrowlong”!

An acre plot was one eighth of a mile long and one eightieth of a mile wide, which is why there are 640 acres in a square mile. Of course once an acre is an area it can be any shape, and it is 4,840 square yards, or 43,560 square feet, which itself is precisely 99% of 44,000 square feet. Since a square rod is known as a perch, an acre is 160 perches. And BTW, the playing area of a standard US football field is roughly 0.9 acres.

Don’t even get me started on Imperial British units for weight, including a hundredweight (which is 112 pounds, of course), one 20th of an Imperial ton (2,240 pounds), and itself four quarters (28 pounds each), or 8 stone (14 pounds each). No, I won’t get started… and certainly not on money made up of pounds, shillings, and pence, with 20 shillings to a pound, and 21 shillings to a guinea for fancy stores, with 12 pence to a shilling, and a half crown was two shillings and six pence, or 2/6 (“two and six”, or 2s 6d). I won’t get started there, either…