Scaling and Order-of-Magnitude

Life would be very different if you were the size of an insect.	Amoebas this size are seldom encountered.

Why can't an insect be the size of a dog? Some skinny stretched-out cells in your spinal cord are a meter tall - why does nature display no single cells that are not just a meter tall, but a meter wide, and a meter thick as well? Believe it or not, these are questions that can be answered fairly easily without knowing much more about physics than you already do. The only mathematical technique you really need is the humble conversion, applied to area and volume.

Area and volume

Area can be defined by saying that we can copy the shape of interest onto graph paper with 1 cm × 1 cm squares and count the number of squares inside. Fractions of squares can be estimated by eye. We then say the area equals the number of squares, in units of square cm. Although this might seem less "pure" than computing areas using formulae like A = πr² for a circle or A = wh/2 for a triangle, those formulae are not useful as definitions of area because they cannot be applied to irregularly shaped areas.

Units of square cm are more commonly written as cm² in science. Of course, the unit of measurement symbolized by "cm" is not an algebra symbol standing for a number that can be literally multiplied by itself. But it is advantageous to write the units of area that way and treat the units as if they were algebra symbols. For instance, if you have a rectangle with an area of 6m² and a width of 2 m, then calculating its length as (6 m²)/(2 m) = 3 m gives a result that makes sense both numerically and in terms of units. This algebra-style treatment of the units also ensures that our methods of converting units work out correctly. For instance, if we accept the fraction

as a valid way of writing the number one, then one times one equals one, so we should also say that one can be represented by

which is the same as

That means the conversion factor from square meters to square centimeters is a factor of 10⁴ , i.e., a square meter has 10⁴ square centimeters in it.

All of the above can be easily applied to volume as well, using one-cubic-centimeter blocks instead of squares on graph paper.

To many people, it seems hard to believe that a square meter equals 10000 square centimeters, or that a cubic meter equals a million cubic centimeters - they think it would make more sense if there were 100 cm² in 1 m², and 100 cm³ in 1 m³, but that would be incorrect. The examples shown in figure b aim to make the correct answer more believable, using the traditional U.S. units of feet and yards. (One foot is 12 inches, and one yard is three feet.)

b / Visualizing conversions of area and volume using traditional U.S. units.

Self-Check	Based on figure b, convince yourself that there are 9 ft2 in a square yard, and 27 ft3 in a cubic yard, then demonstrate the same thing symbolically (i.e., with the method using fractions that equal one).
Answer	1 yd2 × (3 ft/1 yd)2 = 9 ft2 1 yd3 × (3 ft/1 yd)3 = 27 ft3

→ Solved problem: converting mm2 to cm2 page 66, problem 10

→ Solved problem: scaling a liter page 67, problem 19

Discussion Question

A	How many square centimeters are there in a square inch? (1 inch = 2.54 cm) First find an approximate answer by making a drawing, then derive the conversion factor more accurately using the symbolic method.