**Square and cubic units**

Two
questions.

We measure area in square inches. Why? Is it because our unit is a
square with sides 1 inch long?

We
measure volume in cubic inches. Why? Is it because the unit is a cube
with edge 1 inch long?

The
answer to both questions is NO.

A square with a one-inch long side is
only one example of a figure with an area of 1 sq. in. Any other figure with the same area would
do. (For example, an isosceles right
triangle with hypotenuse two inches is such a figure. See the diagram)

Similarly,
one cubic inch of clay that is shaped into a ball or even into a small animal
could serve as an example of an object that has a volume of 1 cu. in., instead
of the traditional cube.

The
shape of the figure that is used as the unit of area doesn't matter. And similarly, the shape of the object that
is used as the unit of volume is also irrelevant.

Additional
comments.

Think
about other common units of area and volume, such as an acre or a gallon.

Having
one acre of land doesn't even suggest its shape. One acre only describes the amount of land
owned.

One
gallon of milk remains 1 gallon, independent of the shape of the
container. And I never saw a 1-gallon
container that was a cube!

So
why do we use square and cubic inches for measuring area and volume?

Consider
any two similar objects, for example, two spheres, and compare their diameters
D_{1} and D_{2}, their surface areas S_{1} and S_{2},
and their volumes V_{1} and V_{2}.

If
the ratio of their diameters is

D_{1}/D_{2} = x

then
the ratio of their surface areas is

S_{1}/S_{2} = x^{2}

and
the ratio of their volumes is

V_{1}/V_{2} = x^{3}

This
means that the relationship between the objects' linear dimensions and their
surface areas is quadratic, and that the relationship between their linear
dimensions and their volumes is cubic. We reflect this fact by naming the units for area and volume *squares* and *cubes* of the linear unit.

This
has many advantages. The main one is
that we can use plain algebra to find out what units to use to measure other
quantities.

For
example, if I measure length in inches, then what unit should I use to measure
the ratio of the volume to the surface area of the object? (The ratio of volume to surface area is
important because it is crucial in computing the cooling rate of physical
objects.)

I measure volume in in^{3}
and area in in^{2}, so their ratio must be measured

in inches, because in^{3}/in^{2}
= in.

Remark.

Using
algebra to handle units in physics is called dimensional analysis.

Composite
units.

Physical
quantities of different kinds cannot be added or subtracted. For example, there is no physical quantity
that is the sum of time and length. But
two quantities of different kinds can be divided or multiplied. This is reflected by the use of composite
units.

In
everyday life, we mostly form simple ratios, such as dollars per pound (unit
cost), or miles per hour (speed). But in
physics and other sciences, some very complex quantities are often studied and
measured in complex units.

Force
in physics is measured in newtons.

one newton = one
kilogram*meter/second^{2}

Does
it mean that one newton is the ratio of a rectangle which has one side that
is 1 kilogram and the other side one
meter, to a square whose two sides are one second? No! Saying that would be pure nonsense!

It
only means that if we would keep the forces (newtons) and the mass
(kilograms) constant, the relationship
between time (seconds) and distance (meters) is quadratic.

Thus
composite units reflect relationships among measured quantities; they do not
reflect relationships about the units themselves.

Webpage
developed by Aous Manshad

Last Modified: August 20, 2009