**Metric Square Units**

The unit which is most often squared or cubed is the length unit. In SI, the base length unit is the meter, m. A square unit, such as m^{2}, describes area and a cubic unit, m^{3}, describes volume.

**Brief Review**

With SI units (like meter, m), prefixes are added to indicate that a fraction (like centimeter, cm or 10¯^{2} m) or a multiple (like kilometer, km or 10^{3} m) is being used.

Suppose we wish to convert a value like 4.90 cm to the equivalent number of km. We need to do two things, (1) determine if we are converting to a bigger unit or smaller and (2) the absolute exponential distance between the two units. Smaller to bigger means a negative sign and the absolute exponential distance is 5. Therefore 4.90 cm = 4.90 x 10¯^{5} km

Or, to use an even simpler example: 1.00 meter would equal 100. cm. The absolute exponential distance here is 2. Keep this problem in mind for the next few paragraphs.

**Converting from one square unit to another**

The big difference in this technique has to do with the exponential distance between two units. You must square the distance to get the right answer.

Let's consider 1.00 m^{2}, one square meter. How many square centimeters are there in this area?

In order to solve this problem, you must ee that there are TWO sides to the square meter area and that EACH SIDE must be converted to cm. Consequently, we have 100.0 (10^{2}) cm on each side and when we multiply them together we get 10^{4} cm^{2}.

Notice that the exponential distance is an exponent distance of 4 (that is, 10^{4}) where as in the meter to cm conversion, it was a distance of 2 (that is, 10^{2})

Let's do a longer distance. Consider 1.00 km^{2}. Convert it to μm^{2} (square micrometers).

Kilo is 10^{3} and micro is 10¯^{6}, so the total absolute exponential distance is 9 (which means 10^{9}). Since we are dealing with a square unit change, we must square 10^{9}, giving 10^{18} as an answer. We are going from one large unit (kilo-) to a smaller unit (micro-), so the sign on the answer will be positive:

1.00 km^{2} = 1.00 x 10^{18} μm^{2}

**Square problems**

1) Convert 4.26 x 10^{4} m^{2} to km^{2}

2) Convert 3.20 x 10^{10} fm^{2} to cm^{2}.

3) Convert the answer in number 2 to Mm^{2}