With the passage of time, which of the following scales will NOT give accurate results due to shrinkage of the sheet or the paper?
i. Plane scale
ii. Engineer’s scale
iii. Representative fraction
iv. Diagonal scale
📝 Detailed Explanation: Graphical vs. Numerical Scales
The core of this question lies in understanding the physical nature of a map and how different types of scales interact with it. Paper is not dimensionally stable; it shrinks or expands over time due to changes in humidity and temperature. This distortion affects the accuracy of measurements.
Scales that Remain Accurate (Graphical Scales)
i. Plain Scale & iv. Diagonal Scale:
Both plain and diagonal scales are graphical scales. This means they are drawn as physical lines or bars directly onto the map. The crucial point is that when the map paper shrinks, these graphical scales shrink at the exact same rate as the rest of the map features. The proportion between the map and its scale is perfectly maintained. Therefore, if you take a measurement from the shrunken map and compare it to the shrunken graphical scale, the reading will still be accurate.
Scales that Become Inaccurate (Numerical/Stated Scales)
ii. Engineer’s Scale & iii. Representative Fraction:
These are numerical or stated scales. They are not graphical representations but are text that states a ratio.
- An Engineer's Scale is a statement like "1 cm = 50 m".
- A Representative Fraction (R.F.) is a ratio like "1:5000".
When the paper shrinks, a line that was originally drawn to be 1 cm long might now measure only 0.98 cm. However, the text on the map still reads "1 cm = 50 m". If you use a ruler to measure the 0.98 cm line and apply the original scale, you'll get an incorrect ground distance. The stated ratio is no longer true for the physically altered drawing. These scales become unreliable because they do not change along with the paper.
⚙️ Formulas & Practical Example
To quantify the error, we use the Shrinkage Factor (SF) or Shrinkage Ratio.
Shrinkage Factor (SF) = Shrunk Length / Original Length
The new, corrected scale after shrinkage is called the Shrunk Scale.
Shrunk Scale = SF × Original Scale (R.F.)
Example Scenario:
- A line on a map was originally drawn to be 20 cm long.
- The map's original R.F. was 1:1000.
- After a few years, the paper shrinks, and the same line now measures 19.6 cm.
- Calculate the Shrinkage Factor (SF):
SF = 19.6 cm / 20 cm = 0.98
- The Problem with the Original R.F.:
The true ground distance represented by the line is: 20 cm × 1000 = 20,000 cm = 200 m.
If a surveyor now measures the 19.6 cm line but uses the original R.F., they get: 19.6 cm × 1000 = 19,600 cm = 196 m. This is an error of 4 meters! - How a Graphical Scale Solves This:
A graphical scale drawn on the map would also have shrunk by the same factor (0.98). If a 10 cm segment of the scale was drawn, it would now be 9.8 cm long. By measuring the 19.6 cm line with dividers and comparing it to the shrunken graphical scale, the surveyor would see it matches two 9.8 cm segments, correctly reading the original ground distance of 200 m.
💡 Key Concepts for Students
- Graphical Scales (Plain, Diagonal): Physically drawn on the map, they shrink with the map and remain accurate.
- Numerical Scales (Engineer's, R.F.): Stated as text or a ratio, they do NOT shrink with the map and become inaccurate.
- Best Practice: For this reason, all professional and legal maps should include a graphical scale to ensure accurate measurements can be made even from a distorted copy.