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.

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 D1 and D2, their surface areas S1 and S2, and their volumes V1 and V2.

If the ratio of their diameters is

D1/D2 = x

then the ratio of their surface areas is

S1/S2 = x2

and the ratio of their volumes is

V1/V2 = x3

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 in3 and area in in2, so their ratio must be measured

in inches, because in3/in2 = 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/second2

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.