36
1 yd
2
x(3 ft/1 yd)
2
=9 ft
2
.1 yd
3
x(3 ft/1 yd)
3
=27 ft
3
.
converting units work out correctly. For instance, if we accept the fraction
100cm
1m
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
100cm
1m
×
100cm
1m
which is the same as
10000cm
2
1m
2
.
That means the conversion factor from square meters to square centi-
meters is a factor of 10
4
, i.e. a square meter has 10
4
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 cm
2
in 1 m
2
, and 100 cm
3
in 1 m
3
, but that would be incorrect. The examples
shown in the figure below 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.)
1 ft
1 yd = 3 ft
1 ft21 yd2 = 9 ft2
1 ft3
1 yd3 = 27 ft3
Self-Check
Based on the figure, convince yourself that there are 9 ft
2
in a square yard ,
and 27 ft
3
in a cubic yard, then demonstrate the same thing symbolically (i.e.
with the method using fractions that equal one).
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.
Chapter 1Scaling and Order-of-Magnitude Estimates