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Chapter 2 Map Scale
What Is Scale
As highlighted in the previous chapter, **scale** is a fundamental requirement for any map. A drawing showing a network of lines and shapes without a scale is considered merely a **sketch**. The scale is crucial because it establishes the quantitative relationship between a map and the real world it represents.
Essentially, a map scale indicates how much the actual ground features have been reduced to be depicted on the map. It represents the ratio between a distance measured on the map and the corresponding distance on the Earth's surface. For instance, if a map has a scale bar, its divisions show that a certain length on the map corresponds to a specific distance (like kilometers or miles) on the ground.
Key terms related to expressing scale as a fraction or ratio:
Glossary Terms:
| Term | Definition |
|---|---|
| Denominator | In a fraction (like 1:50,000), this is the number below the line (e.g., 50,000). It represents the number of units on the ground. |
| Numerator | In a fraction (like 1:50,000), this is the number above the line (e.g., 1). It represents one unit of distance on the map. |
| Representative Fraction (R. F.) | A way to express map scale as a fraction or ratio (e.g., 1/50,000 or 1:50,000) that shows the relationship between one unit of distance on the map and the equivalent distance in the same units on the ground. |
Methods Of Scale
There are primarily three standard methods used to express the scale of a map. Each method has its own way of representing the map-to-ground relationship and comes with specific advantages and limitations.
Before discussing the methods, it's important to be aware of the different systems of linear measurement commonly used. The **Metric System** (using kilometers, meters, centimeters, millimeters) is used in India and many other countries, while the **English System** (using miles, furlongs, yards, feet, inches) is prevalent in the United States and the United Kingdom. India used the English system before adopting the metric system in 1957.
Systems of Measurement (Box 2.1):
| 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 |
The three methods for showing scale on a map are:
Statement Of Scale
In this method, the scale is expressed in words, as a written statement directly on the map. For example, "1 centimetre represents 10 kilometres" or "1 inch represents 10 miles". This means that every centimetre measured on the map corresponds to an actual distance of 10 kilometres on the ground, or every inch on the map corresponds to 10 miles on the ground.
Advantages:
- It is easy to understand for anyone familiar with the units mentioned.
Limitations:
- It is not universally understandable; someone familiar with the metric system may not easily grasp a scale given in miles and vice-versa.
- If the map is photographically enlarged or reduced, the statement of scale becomes incorrect and misleading, requiring a new scale to be calculated and written.
Graphical Or Bar Scale
This method uses a visual representation of the map scale. It involves drawing a straight line or bar, which is divided into segments. These segments are marked with the actual ground distances they represent (e.g., 0 km, 1 km, 2 km, etc.). The leftmost segment is often further subdivided into smaller units for more precise measurement (e.g., in meters or furlongs).
Advantages:
- Its major advantage is that the scale remains correct even if the map is enlarged or reduced photographically. Since the bar scale is part of the map image, it gets enlarged or reduced along with the map, maintaining the correct ratio between map distance (on the bar) and ground distance (marked on the bar).
- It is intuitive to use for direct measurement on the map.
Limitations:
- Like the statement scale, the readings on the bar scale are in a specific system of measurement (metric or English), which may not be universally understood.
Representative Fraction (R. F.)
The **Representative Fraction (R. F.)** expresses the map scale as a ratio or fraction. It shows the relationship between map distance and ground distance as a ratio of units of length, where the numerator is always 1. For example, an R. F. of 1:50,000 means that 1 unit of distance on the map represents 50,000 of the *same units* of distance on the ground. This could mean 1 centimeter on the map represents 50,000 centimeters on the ground, or 1 inch on the map represents 50,000 inches on the ground, or 1 millimeter on the map represents 50,000 millimeters on the ground, and so on.
Advantages:
- It is the **most versatile and universally applicable** method of scale. Since it is expressed as a unitless ratio, users can interpret it using any unit of measurement they are familiar with (centimeters, inches, meters, feet, etc.). This makes it internationally standard.
Limitations:
- It requires calculation to convert the ground units into more familiar larger units (like km or miles) or to draw a graphical scale from it.
- Like the statement scale, if the map is enlarged or reduced, the stated R. F. becomes incorrect (unless a new R. F. is calculated).
Conversion Of Scale
Being able to convert a scale from one method to another is a useful skill in map work. Conversions are often necessary to use a map effectively, especially when working with different systems of measurement.
Statement Of Scale Into R. F.
To convert a Statement of Scale into an R. F., both the map distance and the ground distance must be expressed in the **same unit**.
Problem. Convert the given Statement of Scale of 1 inch represents 4 miles into R. F.
Answer:
The given statement is: 1 inch represents 4 miles.
We need to express both sides in the same unit. Let's convert miles to inches using the conversion: 1 mile = 8 furlongs, 1 furlong = 220 yards, 1 yard = 3 feet, 1 foot = 12 inches.
1 mile = 8 furlongs $\times$ 220 yards/furlong = 1760 yards
1760 yards $\times$ 3 feet/yard = 5280 feet
5280 feet $\times$ 12 inches/foot = 63,360 inches
So, 1 mile = 63,360 inches.
Now, substitute this into the statement of scale:
1 inch represents 4 $\times$ 63,360 inches
1 inch represents 253,440 inches
Since both map distance (1 inch) and ground distance (253,440 inches) are in the same unit, we can express this as a unitless ratio, which is the R. F. We can replace "inch" with "unit".
1 unit represents 253,440 units
This is written as a fraction or ratio:
Answer: R. F. = 1 : 253,440
R. F. Into Statement Of Scale
To convert an R. F. into a Statement of Scale, we interpret the R. F. using a specific unit for both the numerator (map distance) and denominator (ground distance), and then convert the ground distance into a more convenient larger unit (like km or miles).
Problem. Convert R. F. 1 : 253,440 into Statement of Scale (In Metric System)
Answer:
The given R. F. is 1 : 253,440.
This means 1 unit on the map represents 253,440 units on the ground.
In the Metric System, let's choose centimeters as the unit for the map distance.
So, 1 cm on the map represents 253,440 cm on the ground.
Now, convert the ground distance from centimeters to kilometers, as kilometers are a more convenient large unit in the metric system. We know that 1 km = 1000 meters, and 1 meter = 100 centimeters. Therefore, 1 km = 1000 $\times$ 100 = 100,000 cm.
To convert centimeters to kilometers, divide by 100,000.
253,440 cm = $\frac{253,440}{100,000}$ km = 2.5344 km
So, 1 cm represents 2.5344 km.
Rounding off to two decimal places as suggested in the text example:
Answer: 1 cm represents 2.53 km
Construction Of The Graphical/Bar Scale
A graphical scale (or bar scale) is constructed as a visual line segment on the map, with divisions marked to represent specific ground distances. Constructing a graphical scale from an R. F. or Statement of Scale involves calculating the length of the line segment on the map that corresponds to a convenient round number of ground units (like kilometers or miles) and then dividing that line segment appropriately.
By convention, a graphical scale is typically drawn with a total length of about 10-15 cm (Metric) or 6 inches (English System) for ease of use and placement on the map.
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. is 1 : 50,000. We need to construct a graphical scale reading in kilometers and meters.
This R. F. means that 1 unit on the map represents 50,000 units on the ground.
Let's use centimeters as the unit on the map (as is convention for Metric systems).
So, 1 cm on the map represents 50,000 cm on the ground.
Convert ground distance to kilometers (1 km = 100,000 cm):
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.
Now, let's determine the length of the scale bar. By convention, we aim for a length of around 15 cm. If 1 cm represents 0.5 km, then 15 cm represents 15 $\times$ 0.5 km = 7.5 km.
Since 7.5 km is not a round number for the total distance represented by the scale bar, we choose a convenient round number close to 7.5 km, such as 5 km (or 10 km).
Let's choose to show a total of 5 km on our graphical scale. We need to calculate the length of the line on the map that represents 5 km on the ground.
If 0.5 km is represented by 1 cm,
Then 5 km is represented by $\frac{5}{0.5}$ cm = 10 cm.
So, a line of 10 cm on the map will represent 5 km on the ground.
Construction Steps:
- Draw a straight line segment exactly 10 cm long.
- Divide this 10 cm line into 5 equal main divisions. Each main division will represent $\frac{5 \text{ km}}{5} = 1 \text{ km}$ on the ground.
- Label the divisions starting from 0 at the beginning of the second main division from the left. Label the points to the right as 1 km, 2 km, 3 km, 4 km. The total length to the right of 0 represents 4 km. The segment to the left of 0 represents 1 km.
- The leftmost segment (from the start of the line to 0) also represents 1 km. Divide this leftmost segment into smaller secondary divisions to show meters. If you divide it into 10 equal parts, each part represents $\frac{1 \text{ km}}{10} = 0.1 \text{ km} = 100 \text{ meters}$.
- Label the subdivisions to the left of 0 as 100m, 200m, 300m, ..., 1000m (or 1 km, which is the main division length). Start labeling from 0.
- The scale bar should look similar to Figure 2.2, with appropriate labels in km and m.
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 need to construct a graphical scale reading in miles and furlongs.
By convention, we aim for a graphical scale length of around 6 inches in the English system.
If 1 inch represents 1 mile, then 6 inches will represent 6 $\times$ 1 mile = 6 miles.
So, a line of 6 inches on the map will represent 6 miles on the ground.
We also need to show furlongs in the secondary divisions. We know that 1 mile = 8 furlongs.
Construction Steps:
- Draw a straight line segment exactly 6 inches long.
- Divide this 6-inch line into 6 equal main divisions. Each main division will represent $\frac{6 \text{ miles}}{6} = 1 \text{ mile}$ on the ground.
- Label the divisions starting from 0 at the beginning of the second main division from the left. Label the points to the right as 1 mile, 2 miles, 3 miles, 4 miles, 5 miles. The total length to the right of 0 represents 5 miles. The segment to the left of 0 represents 1 mile.
- The leftmost segment (from the start of the line to 0) also represents 1 mile. Divide this leftmost segment into smaller secondary divisions to show furlongs. Since 1 mile = 8 furlongs, we can divide this segment into 8 equal parts. Each part represents $\frac{1 \text{ mile}}{8} = 0.125 \text{ miles} = 1 \text{ furlong}$.
- Label the subdivisions to the left of 0 as 2, 4, 6, 8 Furlongs (or mark every other furlong). The text example (Figure 2.3) shows labeling every 2 furlongs.
- The scale bar should look similar to Figure 2.3, with appropriate labels in miles and furlongs.
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. is 1 : 50,000. We need to construct a graphical scale reading in miles and furlongs.
This R. F. means that 1 unit on the map represents 50,000 units on the ground.
Let's use inches as the unit on the map (as is convention for English systems).
So, 1 inch on the map represents 50,000 inches on the ground.
Convert ground distance from inches to miles (1 mile = 63,360 inches):
50,000 inches = $\frac{50,000}{63,360}$ miles $\approx$ 0.7891 miles
So, 1 inch on the map represents approximately 0.7891 miles on the ground.
Now, let's determine the length of the scale bar. By convention, we aim for a length of around 6 inches. If 1 inch represents 0.7891 miles, then 6 inches represents 6 $\times$ 0.7891 miles $\approx$ 4.73 miles.
Since 4.73 miles is not a convenient round number for the total distance represented by the scale bar, we choose a convenient round number close to 4.73 miles, such as 5 miles.
Let's choose to show a total of 5 miles on our graphical scale. We need to calculate the length of the line on the map that represents 5 miles on the ground.
If 0.7891 miles is represented by 1 inch,
Then 5 miles is represented by $\frac{5}{0.7891}$ inches $\approx$ 6.336 inches.
Rounding to two decimal places, we get approximately 6.34 inches. So, a line of 6.34 inches on the map will represent 5 miles on the ground.
Now we need to divide this 6.34-inch line into 5 equal parts. Since 6.34 is not easily divisible by 5 using standard rulers, we can use a geometric method to divide the line into equal segments, as described in the text and shown in Figure 2.4. (The procedure involves drawing angled lines and parallel dotted lines).
Each main division will represent $\frac{5 \text{ miles}}{5} = 1 \text{ mile}$.
The leftmost segment (representing 1 mile) needs to be divided into furlongs (1 mile = 8 furlongs). We can divide this segment into 4 or 8 equal parts. Dividing into 4 parts means each part represents $\frac{1 \text{ mile}}{4} = \frac{8 \text{ furlongs}}{4} = 2 \text{ furlongs}$. Dividing into 8 parts means each part represents 1 furlong.
Construction Steps:
- Draw a straight line segment exactly 6.34 cm long.
- Use a geometric method (like drawing angled lines from the ends and dividing them into 5 equal parts, then connecting the division points across the main line as shown in Figure 2.4) to divide this 6.34 cm line into 5 equal main divisions.
- Label the divisions starting from 0 at the beginning of the second main division from the left. Label the points to the right as 1 mile, 2 miles, 3 miles, 4 miles. The total length to the right of 0 represents 4 miles. The segment to the left of 0 represents 1 mile.
- Divide the leftmost segment (from the start of the line to 0) into 4 equal secondary divisions (each representing 2 furlongs). Label these subdivisions to the left of 0 as 2, 4, 6, 8 Furlongs.
- The scale bar should visually represent 5 miles, with primary divisions in miles and secondary divisions in furlongs.
Exercise
(Exercise questions are not included as per instructions.)