Some of the most common compound units are those of area and volume. Area is a measure of how much two-dimensional space a surface takes up. Look at two of the more common formulas for area:
Area of a circle = (pi) X radius X radius
Area of a rectangle = length X width
Both of these formulas involve multiplying a width times a width. We would therefore expect the units to be units of length times units of length: meters X meters or feet X feet. We use exponential shorthand in this case, turning the units into square meters (abbreviated m2) and square feet (abbreviated ft2)
Where area is two-dimensional, volume is three dimensional, being the amount of space in three dimensions that an object takes up. Consider the following formulas:
Volume of a sphere = (4/3) X (pi) X radius X radius X radius
Volume of a cube = side X side X side
Note that these formulas all involve multiplying a distance by itself three times, a process called cubing. Thus, the units of volume are called cubic meters (meters X meters X meters, or m3) and cubic feet (feet X feet X feet, or ft3).
There are also some units of area and volume with special names. In the English system, a popular unit of area is the acre:
1 acre = 43,560 square feet
In the metric system, a popular unit of volume is the liter. A liter is defined as the volukme of a cube 10 centimeters on a side. This is rougly equivalent to 0.264 gallons, or about 1.06 quarts in English units. How many cubic centimeters is that?
1 liter = (10 cm) X (10 cm) X (10 cm) = 1000 cm3
Now, how many cubic meters is that? In converting units of area and volume, people often make a big mistake.
Don't Let This Happen to You!
A very common mistake is to say that, since there are 100 centimeters in a meter, then there must be 100 cubic centimeters in a cubic meter. Let's see how this works:
1 liter = 1000 cm3 X (1 m3/100 cm3) = 10 m3
Since a cubic meter is the volume of a cube 1 meter on a side, and a meter is just a little more than a yard, then it should be obvious that there is no way that a liter (just a little more than a quart, remember) is the same as TEN cubic meters!
This example illustrates the dangers of trying to convert units after using an equation. As a general rule of thumb,
always convert units before plugging in to an equation!
Now let's do it the right way:
10 cm X (1 meter/100 cm) = 0.1 m
1 liter = (0.1 m) X (0.1 m) X (0.1 m) = 0.001 m3
Now answer the following:
A room has a floor that measures 20 feet by 30 feet. How many square feet of carpeting is needed to cover the floor? How many square yards?
How many cubic centimeters of water will fit into a spherical glass container with a radius of 3 meters?
If someone has a square piece of land 40 acres in area, how long does each side of the land measure, in feet?
In Europe, you order spring water, and the waiter brings out a 5 deciliter container. How many cubic centimeters does that container hold? How many quarts? If there are 16 fluid ounces in a quart, how many ounces of water are you getting?
How many cubic centimeters are in a cubic meter?
Now move on to check your answers and find out about more compound units.
Questions? Ask the designer, Jim Heath
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