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Chapter 2 Map Scale
As learned in the previous chapter, scale is a fundamental requirement for any drawing to be considered a map; without a scale, it is merely a sketch. The map scale is a crucial element because it quantifies the degree of reduction used when representing the Earth's surface on a map. It establishes the relationship between distances measured on the map and the corresponding actual distances on the ground.
This relationship is typically expressed as a ratio, comparing the distance between two points on the map to the real-world distance between those same two points on the Earth's surface. For example, a scale of 1 cm represents 10 km means that every one centimeter on the map corresponds to ten kilometers on the ground.
In a mathematical expression of scale as a fraction (e.g., $1:50,000$), the number above the line is the numerator, representing a unit distance on the map. The number below the line is the denominator, representing the equivalent distance on the ground in the same units.
There are primarily three common ways to express the scale on a map:
- Statement of Scale
- Representative Fraction (R. F.)
- Graphical Scale (or Bar Scale)
Understanding these methods requires familiarity with systems of measurement. Different countries use different systems for measuring linear distances. The Metric System (kilometres, metres, centimeters, millimeters) is widely used, including in India presently. The English System (miles, furlongs, yards, feet, inches) is prevalent in countries like the United States and the United Kingdom and was used in India before 1957. The units of measurement in these systems are provided below:
| Metric System | English System |
|---|---|
| 1 km = 1000 Metres | 1 Mile = 8 Furlongs |
| 1 Metre = 100 Centimetres | 1 Furlong = 220 Yards |
| 1 Centimetre = 10 Millimetres | 1 Yard = 3 feet |
| 1 Foot = 12 Inches |
Methods Of Scale
The scale of a map can be expressed using one or a combination of the methods described below:
Statement Of Scale
This method expresses the map scale in words, stating the relationship between a unit of distance on the map and the corresponding unit of distance on the ground. For example, a statement of scale might read: "1 cm represents 10 km." This means that every 1 centimeter measured on the map corresponds to an actual distance of 10 kilometers on the Earth's surface.
The statement can be in Metric units (e.g., 1 cm to 10 km) or English units (e.g., 1 inch to 10 miles). This is the simplest way to express scale, but it has limitations. Its main disadvantage is that it is only understandable to people familiar with the specified units of measurement. Furthermore, if the map is enlarged or reduced (e.g., photocopied at a different percentage), the original statement of scale becomes inaccurate and a new scale calculation is required.
Graphical Or Bar Scale
A graphical scale, also known as a bar scale, represents the map scale visually using a line or bar divided into segments (primary divisions) that represent specific distances on the ground. Often, the leftmost segment (secondary division) is further subdivided to allow for more precise measurements (Figure 2.1 shows a graphical scale). The scale bar itself is printed on the map.
Illustration of a graphical scale bar with marked distances in kilometres and metres, used for direct measurement on the map.
The readings on a graphical scale are typically shown in a specific system of measurement (e.g., kilometers and meters, or miles and furlongs), which can limit its immediate understanding to users familiar with that system. However, a significant advantage of the graphical scale is that it remains accurate even if the map is enlarged or reduced (e.g., during photocopying), because the bar itself is also enlarged or reduced proportionally along with the map.
Representative Fraction ( R. F. )
The Representative Fraction (R. F.) expresses the map scale as a ratio or a fraction. It shows the relationship between a unit of distance on the map and the corresponding distance on the ground in the *same units of length*. R. F. is typically written as a fraction (e.g., $1/50,000$) or as a ratio (e.g., $1:50,000$).
In the R. F. expression $1:50,000$, the numerator (1) represents one unit of length on the map, and the denominator ($50,000$) represents $50,000$ units of the same length on the ground. This means that 1 millimeter on the map represents $50,000$ millimeters on the ground, 1 centimeter represents $50,000$ centimeters, and 1 inch represents $50,000$ inches.
The R. F. is considered the most versatile and universally applicable method of expressing map scale because it is unitless. A user can interpret the R. F. using any unit of measurement they are familiar with. For example, with an R. F. of $1:36,000$:
- A user familiar with the Metric System can interpret it as 1 cm on the map representing $36,000$ cm on the ground. Converting to meters ($36,000 \, cm / 100 \, cm/m = 360 \, m$), the scale is 1 cm represents 360 meters.
- A user familiar with the English System can interpret it as 1 inch on the map representing $36,000$ inches on the ground. Converting to yards ($36,000 \, inches / 36 \, inches/yard = 1,000 \, yards$), the scale is 1 inch represents 1,000 yards.
This ability to use any unit makes R. F. a universal method, easily understood and converted by users regardless of their preferred measurement system.
Conversion Of Scale
It is often necessary to convert a map scale from one method of expression to another. This requires knowing the relationships between units in the measurement systems (see Box 2.1 table above).
Statement Of Scale Into R. F.
To convert a Statement of Scale into Representative Fraction, the key is to express both the map distance and the ground distance in the same unit and then write it as a ratio with the map distance as 1.
Problem 1. Convert the given Statement of Scale of 1 inch represents 4 miles into R. F.
Answer:
The given Statement of Scale is: 1 inch represents 4 miles
To convert this into R. F., we need to express both map distance (1 inch) and ground distance (4 miles) in the same unit, which is inches in this case.
We know that 1 mile = 63,360 inches.
So, 4 miles = $4 \times 63,360$ inches = $253,440$ inches.
Thus, the statement "1 inch represents 4 miles" is equivalent to "1 inch represents $253,440$ inches".
Now, we can express this as a ratio with the map distance as the numerator (1) and the ground distance as the denominator:
$R. F. = \frac{\text{Map distance}}{\text{Ground distance}} = \frac{1 \text{ inch}}{253,440 \text{ inches}}$
Since the units are the same, we can write it as a unitless ratio:
$R. F. = 1 : 253,440$
R. F. Into Statement Of Scale
To convert Representative Fraction into Statement of Scale, you need to choose a unit for the numerator (e.g., cm or inch) and then convert the denominator (which is in the same unit) into a more convenient unit for ground distance (e.g., km or miles).
Problem 1. Convert R. F. 1 : 253, 440 into Statement of Scale (In Metric System)
Answer:
The given R. F. is 1 : 253, 440.
This means that 1 unit on the map represents 253,440 units on the ground.
In the Metric System, we usually start with 1 cm as the unit for map distance.
So, 1 cm on the map represents 253,440 cm on the ground.
To convert the ground distance from centimeters to kilometers, we use the relationship: 1 km = 100,000 cm.
Ground distance in km = $\frac{253,440 \, \text{cm}}{100,000 \, \text{cm/km}} = 2.5344 \, \text{km}$.
Rounding off to two decimal places, we get 2.53 km.
Therefore, the Statement of Scale in the Metric System is: 1 cm represents 2.53 km.
Construction Of The Graphical/bar Scale
To construct a graphical scale, you need to determine the length of the line that will represent a convenient round number of ground units (e.g., 1 km, 5 km, 1 mile). By convention, the graphical scale bar is typically drawn to a length of around 10-15 cm or 6 inches, depending on the measurement system.
Problem 1. Construct a graphical scale for a map drawn at a scale of 1 : 50,000 and read the distances in kilometre and metre.
Answer:
Given R. F. = 1 : 50,000.
We want to read distances in kilometers and meters.
First, determine the length of the line to represent a certain distance. By convention, let's aim for a line length of about 10-15 cm.
From the R. F., 1 cm on the map represents 50,000 cm on the ground.
Convert ground distance to kilometers: 50,000 cm = $\frac{50,000}{100,000}$ km = 0.5 km.
So, 1 cm on the map represents 0.5 km on the ground.
Let's choose a convenient length for our scale bar, say 10 cm (within the 10-15 cm range). The ground distance represented by 10 cm will be $10 \times 0.5$ km = 5 km.
So, a line of 10 cm on the map will represent 5 km on the ground.
Now, we construct the graphical scale:
Draw a straight line 10 cm long.
Divide this line into 5 equal primary divisions (each representing 1 km). Mark the divisions as 0, 1, 2, 3, 4 km to the right of the 0 mark.
Divide the leftmost primary division (from 0 to the left) into 10 equal secondary divisions. Since this section represents 1 km, each subdivision will represent $\frac{1}{10}$ km = 0.1 km, or 100 meters (1 km = 1000 m, so 0.1 km = 100 m).
Label these subdivisions to the left of 0 as 100 m, 200 m, ..., 1000 m (or 1 km). You can choose to label only a few subdivisions (e.g., 0, 500 m, 1000 m) for clarity, or label every 100 m.
Diagram illustrating the construction of a graphical scale bar for a map with an R.F. of 1:50,000, showing primary divisions in kilometres and secondary divisions in meters to the left of the zero mark.
Problem 2. Construct a graphical scale when the given Statement of Scale is 1 inch representing 1 mile and read the distances in miles and furlongs.
Answer:
Given Statement of Scale: 1 inch represents 1 mile.
We want to read distances in miles and furlongs.
By convention, let's choose a scale bar length of about 6 inches.
If 1 inch represents 1 mile, then 6 inches will represent $6 \times 1$ mile = 6 miles.
Now, we construct the graphical scale:
Draw a straight line 6 inches long.
Divide this line into 6 equal primary divisions. Assign a value of 1 mile to each division. Mark the divisions as 0, 1, 2, 3, 4, 5 miles to the right of the 0 mark.
We need to show furlongs in the secondary division (left of 0). We know 1 mile = 8 furlongs. The leftmost section (from 0 to the left) represents 1 mile.
Divide the leftmost primary division into 4 equal secondary divisions. Since 1 mile = 8 furlongs, 4 subdivisions will represent $\frac{8}{4} = 2$ furlongs each.
Mark the subdivisions to the left of 0 as 2, 4, 6, 8 furlongs.
Diagram illustrating the construction of a graphical scale bar for a map with a scale of 1 inch representing 1 mile, showing primary divisions in miles and secondary divisions in furlongs to the left of the zero mark.
Problem 3. Construct a graphical scale when the given R. F. is 1 : 50,000 and read the distances in miles and furlongs.
Answer:
Given R. F. = 1 : 50,000.
We want to read distances in miles and furlongs (English System).
By convention, let's choose a scale bar length of about 6 inches.
From the R. F., 1 inch on the map represents 50,000 inches on the ground.
Convert ground distance from inches to miles: We know 1 mile = 63,360 inches.
So, 50,000 inches = $\frac{50,000}{63,360}$ miles $\approx 0.7891$ miles.
So, 1 inch represents $\approx 0.7891$ miles.
A line length of 6 inches will represent $6 \times 0.7891$ miles $\approx 4.7346$ miles.
The value 4.7346 miles is not a convenient round number. Let's choose a round number of ground miles that will be represented by our scale bar. A number close to 4.73 miles, such as 4 miles or 5 miles, would be suitable. Let's choose 5 miles to be represented by the scale bar.
Now, determine the map length required to represent 5 miles:
We know 1 inch represents 0.7891 miles (approximately).
To find the length representing 5 miles, we can set up a proportion or use the R.F.:
$R. F. = \frac{\text{Map Distance (inches)}}{\text{Ground Distance (inches)}}$
$\frac{1}{50,000} = \frac{\text{Map Length (inches)}}{5 \text{ miles}}$
Convert 5 miles to inches: 5 miles = $5 \times 63,360$ inches = $316,800$ inches.
Map Length (inches) = $\frac{1}{50,000} \times 316,800$ inches = $\frac{316,800}{50,000}$ inches = 6.336 inches.
So, a line of 6.336 inches on the map will represent 5 miles on the ground. This is a convenient length for a graphical scale (close to our initial aim of 6 inches).
Now, we construct the graphical scale:
Draw a straight line 6.336 inches long.
Divide this line into 5 equal primary divisions (each representing 1 mile). Mark the divisions as 0, 1, 2, 3, 4 miles to the right of the 0 mark.
We need to show furlongs in the secondary division (left of 0). We know 1 mile = 8 furlongs.
Divide the leftmost primary division (from 0 to the left, representing 1 mile) into 4 equal secondary divisions. Since 1 mile = 8 furlongs, 4 subdivisions will represent $\frac{8}{4} = 2$ furlongs each.
Mark the subdivisions to the left of 0 as 2, 4, 6, 8 furlongs.
To divide a line segment of length L into 'n' equal parts using the method shown in Figure 2.4:
- Draw the line segment of length L (here, 6.336 inches).
- From one endpoint (say, the left end), draw a line at a convenient acute angle (e.g., $40^\circ$ or $45^\circ$).
- Mark 'n' equally spaced points along this angled line (e.g., 5 points for dividing into 5 parts, spacing of 1 or 1.5 inches suggested).
- From the other endpoint of the main line segment (the right end), draw a second line at the same acute angle but in the opposite direction (or downward).
- Mark 'n' equally spaced points along this second angled line, starting from the endpoint and moving outwards with the same spacing used for the first angled line.
- Join the corresponding points on the two angled lines with dotted lines (e.g., 1st point on line 1 joined to 1st point on line 2, 2nd to 2nd, etc.).
- The points where these dotted lines intersect the original line segment of length L will divide it into 'n' equal parts.
Illustration of a geometrical method for dividing a line segment into a specific number of equal parts, used in constructing graphical scales.
Exercise
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