This handout was designed to help you understand the relationships between different map scales, map units, distance, and area. You should understand the logic behind how these are used and how they should appear on real topographic maps. On your upcoming lab exam, you should also be able to convert one type of map scale into another, and calculate size differences (scale factor, area factor) between different maps.

All map scales are an expression of the numerical relationship between the MAP and the LAND that is represented. The MAP unit is always mentioned first.

1. Verbal Scale: The verbal scale is just a sentence stating that "1 Map Unit = X Land Units". For reasons of convenience, a mixture of units is commonly used, such as

However, there are NO requirements that the units must be different! The expression "1 inch = 63,360 inches" is still a verbal scale. A mixture of map and land units makes the verbal scale difficult to compare between different maps - it must be converted first to a Representative Fraction (see below).

which means "1 map unit is equivalent to 24,000 land units." Because an R.F. carries no units (inches, centimeters, etc.), it means that the R.F. scales can be compared between different maps. Converting an R.F. scale to a verbal scale is very easy; simply select ONE unit and apply it to BOTH map and land numbers. The above example can be written as a verbal scale as "1 inch = 24,000 inches" or "1 meter = 24,000 meters," etc. (Note: YOU CANNOT MIX UNITS in an R.F.! Doing so will change the numerical relationship of the R.F.)

When converting a verbal scale to an R.F., the strategy is to convert from mixed units (verbal scale) to one unit (R.F.). That is the basic difference between these two types of map scales.

Example:

If your verbal scale is "1 inch = 1 mile" how is this expressed as an R.F.?

Solution:

1. Decide which ONE unit to convert to: To become an R.F., both the map and land units (now 2 different types) must be the same. You have 2 choices to choose from: you can either convert miles to inches or inches to miles. It is usually easier to convert from a larger to a smaller unit ("how many inches are in a mile?" is easier to handle than "how many miles are in an inch?"). So, we will then convert the "1 mile of land" to "X number of inches".

2. Eliminate the unwanted unit by multiplication: One of the basic rules of algebra is that any number or unit divided by itself equals 1. If you started with miles and wanted to get rid of miles and end up with inches, how do you do this?

First, get rid of "miles" by multiplying it by a fraction that contains "miles" in the denominator, and an equivalent number of smaller units in the numerator. You may not know how many inches there are in a mile, but you should know that there are 5,280 feet in a mile. This will get rid of miles, but will leave you with "feet" which is still not the same unit as the Map Unit (inches). To get to inches, get rid of "feet" by multiplying by a fraction that contains "feet" in the denominator and the equivalent number of inches in the numerator:
 

Now that the original land unit "1 mile" has been converted to 63,630 inches, both the map and land units are now the same type, and the R.F. is written simply by deleting the units and substituting a colon for the equal sign:

There is no "right" way or "wrong" way to multiply - you must decide how to set up the fractions so that the units you don't want get canceled, and the unit you do want ends up as your answer.

C. CONVERTING AN R.F. TO A VERBAL SCALE

Converting an R.F. to a verbal scale is usually much easier than the reverse. By definition, an R.F. means that both the map and land units are the same, so you can choose any ONE unit: 1 : 24,000 can be "1 cm = 24,000 cm" or "1 inch = 24,000," so long as you do not use two different units. Remember, there is NO REQUIREMENT that a verbal scale must use different units!

D. COMPARING SCALES AND AREAS BETWEEN DIFFERENT MAPS

One of the major advantages of using the R.F. (Representative Fraction) scale is that it allows you to directly compare the sizes of objects between different maps. Because the R.F. eliminates the use of specific units (such as inches, feet, miles, etc.), there are no complicated conversions needed.

Comparing R.F. Scales Between Different Maps

Simply divide the larger map scale by the smaller one to get the SCALE FACTOR:
 

Relationship Between Scale & Area

A map scale measures distance, which is a one-dimensional unit. Area is a 2-dimensional quantity, calculated by measuring "Length X Width." Note that when a map scale is changed by a certain number factor, the area changes by the square of that number. In other words, if the scale is 2 times larger, the area becomes 4 times larger; if the scale is 5 times larger, the area becomes 25 times larger:
 

Note that although both maps are the same size, Map A covers 500 units of distance versus only 100 units for Map B. But, Map A (500x500 = 250,000 sq. ft.) covers 25 TIMES the AREA of Map B (100x100 = 10,000 sq. ft/).

So, the relationship between the SCALE FACTOR and AREA FACTOR is: