The fundamental SI units cover all of the basic measuring needs. There are times, however, when it's necessary to relate measurements mathematically. For example, let's say you measure the length of a soccer field and find it to be 120 meters (394 feet) long. Then you determine its width to be 90 meters (295 feet). If you wanted to find the area of the field, you would need to multiply its length by its width. But you don't just multiply the numbers in front of the units; you multiply the units, too. So, the math would look like this:
area = length × width = 120 m × 90 m = 10,800 m2
Notice that the final unit is a meter times a meter, which results in what metrologists, or measuring experts, call a square meter.
Now let's say you have a cube measuring 1 meter on each side. If you wanted to find the volume of the cube, you would need to multiply three dimensions -- length, width and height. Here's the math:
volume = length × width × height = 1 m × 1 m × 1 m = 1 m3 = m3
Notice again that the base unit gets multiplied along with the numerical factor. In this case, it's a meter times a meter times a meter, resulting in a cubic meter. Also observe that when the numerical factor is 1, you can drop the number and simply show the unit. Metrologists call this a coherent unit.
Area and volume are derived units because they are defined in terms of an SI base unit and a specific quantity equation. The table lists some of the most common derived units.
A few derived units are significant enough to have earned special SI names and symbols. Force serves as a great example. Isaac Newton defined force as the mass of an object times its acceleration. When you multiply these two quantities together, you get a derived unit of kilogram meter per second squared (kg-m/s2). Because kg-m/s2 is a little cumbersome and because force is such an important quantity in physics, SI bigwigs decided to call the derived unit a newton, in honor of Sir Isaac. In all, there are 22 derived SI units with special names and symbols. Some of the most important ones appear in the accompanying table.
Finally, it's important to know that a few units are not officially part of the metric system but make frequent appearances. As such, the SI accepts these units for use with its family of measures. Some of the common time quantities -- the minute, hour and day -- fall into this category, as do the metric ton and astronomical unit. All of these units, however, can be defined according to SI base units. For example, a day is 86,400 seconds. And an astronomical unit (AU) -- a unit of length equal to the mean distance between the Earth and the sun -- is 1.495978 × 1011 meters.
Of course, a base unit may be too large or too small to describe an object adequately. In the SI, making units larger and smaller requires nothing more than adding a prefix. We'll cover those on the next page.