Converting Between Scale Types of Historic & Modern Maps, page 3

Converting Lexical Scale to Representative Fraction (RF)/Ratio Scale

English and Metric Measures of Length to use:

1 inch (in) = 2.54 centimeters
1 centimeter (cm) = 0.39 inch
1 foot (ft) = 12 inches = 30.5 centimeters = 0.3 meter
1 yard (yd) = 36 inches = 3 feet = 91.44 centimeters = 0.9 meter
1 meter (m) = 100 centimeters = 39.37 inches = 3.28 feet = 1.09 yards
1 mile (mi) = 63,360 inches = 5,280 feet = 1,760 yards = 1.609 kilometers
1 nautical mile = 1,852 meters = 1.852 kilometers
1 kilometer (km) = 1,000 m = 100,000 cm = 1,094 yards = 0.6 mile

NOTE: When you carry out any calculation with fractions, make sure that all units in equation are cancelled correctly. If necessary, invert the RF for that purpose.

1) IN ENGLISH UNITS

Converting Lexical Scale "1 inch represents 3.9 miles" to Representative Fraction (RF) & Ratio:

First, I write the lexical scale as a fraction - 1 inch (on the map) / 3.9 miles (on the ground), then I convert the length unit mile to inches because the RF shows how many inches on the ground are represented by 1 inch on a map.

Equation for Converting Lexical Scale to RF

I should round the denominator of the resulting fraction so that it does not imply more accuracy than the original precision warranted. So the resulting fraction is 1/250,000 corresponding to a ratio 1:250,000.

Because 1 mile = 5,280 feet x 12 inches = 63,360 inches, I can write the formula for converting Lexical Scale to the RF/Ratio Scale in English units this way:

if Lexical Scale = "1 inch represents A (miles)", then:

Formula for Converting Lexical Scale to RF

where: A (miles) - distance on the ground represented by 1 inch on a map.

2) IN METRIC UNITS

Converting Lexical Scale "6 cm represent 1.5 km" to RF/Ratio:

As in a previous case, I express the lexical expression as a fraction - "6 cm / 1.5 km" and convert kilometres to centimeters, so that the numerator has a "1" in a resulting fraction, and both numerator and denominator have the same units:

Equation for Converting Lexical Scale to RF in centimeters

By solving this equation, I converted a lexical expression "6 cm represent 1.5 km" to a scale as a RF - 1/25,000, or as a Ratio - 1:25,000.

Knowing that 1 km = 100,000 centimetres, I can express the formula for determining the map scale from just a verbal expression in metric units:

if Lexical Scale = "B (cm) represent C (km)", then:

Formula for Converting Lexical Scale to RF in centimeters

and:

Formula for Converting Lexical Scale to RF in centimeters

where:
RF - Representative Fraction Scale of the map,
A (cm) - actual distance - a number of centimetres on the ground, represented by 1 centimetre on a map,
B (cm) - distance on a map,
C (km) - distance on the ground represented by "B" on a map.

And the formula for converting Lexical Scale "1 cm represents A km" to RF & Ratio would look like this:

if "1 cm represents A (km)", then:

Formula for Converting Lexical Scale to RF in Kilometers

where A (km) - distance on the ground represented by 1 cm on a map.


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