Except for Suvrat, those are all examples of unit analysis. A dimension is something that can be measured, like length, or mass, whereas a unit is a standardized scale which is used to measure it. Unit analysis is a wonderful technique for simplifying equations, as well as double checking your work, to make sure you haven't made a mistake. If the units don't match up, you've either used the wrong equation, or dropped a term in your calculation.

Dimensional analysis, however, is for deciding which basic dimensions a unit actually measures, which is very useful to develop unit conversions, for use in unit analysis. There are many units which don't measure basic dimensions, but rather apply to derived dimensions, like pressure, energy, temperature, etc.

A Newton, for example, is a unit to measure the dimension of force, but what is force?

Let's start with the well known equation: force = mass x acceleration

Mass is a basic dimension, but acceleration is not - it is derived from the rate of change of a velocity, or velocity / time, but velocity itself is the rate of change in distance, or length / time

Using only basic dimensions, we've now got

force = mass x length / time2

Applying this to unit analysis, we know that if the unit for force is the Newton, the units on the right better be

kilogram x meter / second2

Similarly, we know that

E = mc2 . So what is energy?

Using the same technique:

  • m = mass
  • c = the velocity of light, which is length / time
  • and so c2 = length2 / time2
So, energy = mass x length2 / time2

If we are using Joules for units of energy, we'd better have kilogram x meter2 / seconds 2 in our result.


Those are some basics - stuff we already know based on standard use of units. Let's try a weird one:

Most people in the US measure their car's use of gas as miles per gallon, (or kilometers per liter outside of the US) and use the unit Mpg for it. Typical small cars get 30 Mpg. But what dimension does a Mpg measure?

Let's start with the dimensional analysis.
A mile is a unit of length, and a gallon is a unit of volume.
Volume is length3.
Hence, we can see that gas efficiency is 1 / length2 , or 1/area.
This is the dimension used for a cross-section.

What on earth does this cross-sectional area, representing gas efficiency refer to? Let's convert Mph (using unit analysis) into standard CGI units, to see how big it is.

1 gallon ~ 4 liters = 4,000 centimeters3
1 mile = 1.6 kilometers = 1,600 meters = 160,000 centimeters

So, 30 miles / gallon
= 30 x 160,000 centimeters / 4,000 centimeters3
= 1,200 / centimeters 2

But what the heck does 1,200 / cm2 mean?

It's a cross-section, so if we invert the value, we'll have an area:

cm 2 / 1,200
= 100 millimeters2 / 1,200
~ 1/12 millimeter2

...which is an extremely small area - it's the area of a square about 1/3 of a millimeter on a side.

OK, great, we now know that 30 Mpg is a cross section of something, and the area of the cross section is tiny. But what is the something, physically? In terms of operating your vehicle, you know that your car's Mpg means that after a certain amount of travel, you have to refill the gas tank, but what does Mpg actually measure which is 1/3 of a millimeter on a side?

You might be tempted to think of the cross section of the fuel line in the engine, which is a pretty good guess, since it's about the same area, but that's not it.

Imagine your car, traveling over a very long, very thin, very short trough, full of gas. As you travel, the car consumes gas at exactly the rate needed to use up the gas, as it goes: Behind you, the trough is empty, in front of you, it's full.

Mpg is the cross sectional area of this trough. That's why it's thin and short:
30 Mph means a trough about 1/3 of a millimeter on a side.
My old Chevy Suburban, with Mph of about 10, would need a trough about 3 times the cross section: about 1/2 millimeter on a side.

Dimensional analysis can also make rotational velocity much easier to understand.