__MAP SCALES & UNITS__

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 inch = 1 mile**

*However, there are NO requirements that the units must be
different!* The expression

2.

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.)*

3.

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

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:

1 mile X |
(5,280 feet) (1 mile) |
X
(12 inches) (1 foot) |
= 63,360 inches |

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:

**1 inch = 63,360 inches**

**1 : 63,360**

*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:

**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:

SCALE FACTOR = |
R.F. of Map A =R.F. of Map B |
1:50,000 1:10,000 |
= 5 |

__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:*

Map A |
Map B |
||||||||||||||||||||||||||

Area = (500 feet horizontal) X (500 feet vertical)
1:50,000.
The above map shows a land area of 500 feet X 500 feet = 250,000 square feet. |
Area = (100 feet horizontal) X (100 feet vertical)
25
times the area of Map B (the square of the scale factor) |

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:

**AREA FACTOR = (SCALE FACTOR) ^{2}**