An example of a physical measurement is “3.0 meters.” The meter is a unit, the type of unit being determined by the dimension to be measured. Meters are units of length, as are kilometers, feet, miles, inches and so on. Whichever of these units is used, length is the dimension being measured. Other fundamental dimensions include time (seconds, hours, days and so on) and mass (kilograms, grams).

Dimensional analysis is a useful tool for analyzing physical situations and checking whether calculations make sense. In dimensional analysis, dimensions are treated algebraically. We use the symbols L, T, and M to represent the dimensions of length, time, and mass. The volume of a cube, for instance, has dimensions L×L×L or L³.

An example will demonstrate the usefulness of dimensional analysis. First, we introduce several terms from the study of motion. The average velocity v of an object is its displacement (the net distance it moves) divided by the time it travels. The dimensions of velocity, then, are L/T. An object's acceleration a, on the other hand, has dimensions L/T².

Let’s say you roughly recall an equation for calculating the distance, x, that an object that starts at rest travels in time t, but cannot remember for sure if it uses velocity or acceleration. But you are sure the equation is either

x = ½vt² or x = ½at²

You can determine the correct equation using dimensional analysis. The left side of the equation, x, represents the distance traveled and has dimension L. So the right side of the equation must also have dimension L. You can then check to see that

dimension of ½vt² = (L/T)(T²) = LT

but

dimension of ½at² = (L/T²)(T²) = L

So the correct form of the equation is the one using the acceleration a.

Notice that we ignored the ½ in the equation. Some quantities used in physics equations are dimensionless, meaning they have no dimension and do not carry units. All pure numbers like ½ or π are dimensionless, as are some measures of angles, like π radians.