Problem Statement
The area of a plot on a map is measured using a planimeter and found to be 10.22 cm². The original scale of the map was 1 : 25,000. However, the map has shrunk over time, such that a line originally 5 cm long now measures only 4.8 cm. What is the correct field area of the plot in hectares?
Step-by-Step Solution
Key Information
- Measured Area on Shrunk Map = 10.22 cm²
- Original Map Scale = 1 : 25,000
- Original Line Length = 5.0 cm
- Current (Shrunk) Line Length = 4.8 cm
- Goal: Find the correct ground area (in hectares).
- Conversion: 1 hectare = 10,000 m²
Step 1: Calculate Shrinkage Factors
The linear shrinkage factor (SF) is the ratio of the shrunk length to the original length.
SF (Linear) = Shrunk Length / Original Length
SF (Linear) = 4.8 cm / 5.0 cm
SF (Linear) = 0.96
The area shrinkage factor is the square of the linear shrinkage factor, as area involves two dimensions.
SF (Area) = (SF (Linear))²
SF (Area) = (0.96)²
SF (Area) = 0.9216
Step 2: Calculate Original Map Area
The area measured on the shrunk map (10.22 cm²) corresponds to the original map area multiplied by the area shrinkage factor.
Measured Shrunk Area = Original Map Area × SF (Area)
To find the original map area before shrinkage:
Original Map Area = Measured Shrunk Area / SF (Area)
Original Map Area = 10.22 cm² / 0.9216
Original Map Area ≈ 11.08941 cm²
Step 3: Calculate Ground Area
First, determine the area scale of the *original* map (1:25,000).
Original Linear Scale: 1 cm = 25,000 cm = 250 m
Original Area Scale: 1 cm² = (250 m) × (250 m)
Original Area Scale = 1 cm² = 62,500 m²
Now, calculate the true ground area using the calculated *original* map area and the original area scale.
Ground Area = Original Map Area (cm²) × Original Area Scale (m²/cm²)
Ground Area ≈ 11.08941 cm² × 62,500 m²/cm²
Ground Area ≈ 693,088.125 m²
Finally, convert the ground area to hectares.
Ground Area (hectares) = Ground Area (m²) / 10,000 m²/hectare
Ground Area (hectares) ≈ 693,088.125 / 10,000
Ground Area ≈ 69.31 hectares
Final Result
Conceptual Explanation & Applications
Core Concepts:
- Map Scale & Area Scale: The stated ratio (1:25,000) defines the relationship between map distance and ground distance (linear scale) and, by squaring, map area and ground area (area scale).
- Map Shrinkage: Physical maps (especially paper) can shrink or expand due to environmental factors (humidity, temperature, age), altering their dimensions and thus invalidating the stated scale for direct measurement.
- Shrinkage Factor: A ratio quantifying the change in dimension (shrunk length / original length). The area shrinkage factor is the square of the linear factor.
- Correction for Shrinkage: Measurements taken on a shrunk map must be corrected to find their equivalent value on the original, unshrunk map before applying the original scale to calculate ground distances or areas. Measured Area = Original Area × Area Shrinkage Factor.
- Planimeter: A mechanical or digital instrument used to measure the area of an irregular shape on a flat surface, such as a plot on a map.
Real-World Applications:
- Historical Land Records: Analyzing old property maps that may have degraded or shrunk over time to determine original boundaries and areas.
- Cartographic Archiving: Assessing the accuracy of historical maps and understanding the potential errors introduced by material instability.
- Surveying and Engineering: When using older plans or blueprints where dimensional stability is uncertain, calculating shrinkage allows for accurate interpretation.
- Resource Management: Recalculating areas from historical aerial photographs or maps that might have distortions due to processing or aging.
- Legal Disputes: Providing evidence regarding original property dimensions when current measurements conflict with historical records affected by shrinkage.
Why It Works:
The planimeter measures the *actual physical area* on the map paper *as it currently exists* (10.22 cm²). However, this paper has shrunk. The shrinkage factor (0.96 linearly, 0.9216 for area) tells us how much smaller the map is compared to its original size. By dividing the measured area by the area shrinkage factor (10.22 / 0.9216), we find the area the plot *would have occupied* on the original, unshrunk map (11.08941 cm²). Since the map’s stated scale (1:25,000) corresponds to its *original* dimensions, we must use this corrected (original) map area with the original area scale (1 cm² = 62,500 m²) to calculate the true ground area accurately. This process effectively reverses the effect of shrinkage before applying the scale conversion.
