The area of the plan of an old map plotted to a scale of 10 meters to 1 cm measures now 100.2 sq2. cm. when measured by a plani meter. The plan is found to have shrunk so that a line originally10 cm long now measures 9.7 cm only. Further, the 20 m chain used is 8cm too short. Find the true area of the survey.

Problem Statement

The area of a plan on an old map, originally plotted to a scale of 10 metres to 1 cm, currently measures 100.2 sq. cm when measured by a planimeter. The plan is found to have shrunk such that a line originally 10 cm long now measures only 9.7 cm. Furthermore, the 20 m chain used for the original survey was actually 8 cm too short. Find the true area of the surveyed field.

Step-by-Step Solution

Key Information

Current Measured Plan Area (A_{plan, current}) = 100.2 cm²
Original Scale = 1 cm : 10 m (Scale Factor S = 10 m/cm)
Original Line Length on Map (L_{map, orig}) = 10 cm
Current Line Length on Map (L_{map, current}) = 9.7 cm
Nominal Chain Length (L_nom) = 20 m
Chain Error = -8 cm = -0.08 m (too short)
Actual Chain Length (L_act) = 20 m – 0.08 m = 19.92 m
Goal: Find the True Field Area (A_true).

Step 1: Calculate Original Plan Area (Correcting for Shrinkage)

First, determine the area the plan *would have* covered before it shrank. Since area shrinks by the square of the linear shrinkage, we adjust the current area.

Original Plan Area (A_{plan, orig}) = A_{plan, current} × (L_{map, orig} / L_{map, current})²

A_{plan, orig} = 100.2 cm² × (10 cm / 9.7 cm)²

A_{plan, orig} ≈ 100.2 cm² × (1.0309)²

A_{plan, orig} ≈ 100.2 cm² × 1.06279

A_{plan, orig} ≈ 106.494 cm²

This represents the area on the map as it was originally drawn.

Step 2: Calculate Measured Field Area (Using Original Scale & Area)

Now, convert the original plan area to the corresponding field area using the original map scale. Remember Area Scale = (Linear Scale Factor)².

Measured Field Area (A_measured) = A_{plan, orig} × S²

A_measured ≈ 106.494 cm² × (10 m / 1 cm)²

A_measured ≈ 106.494 cm² × 100 m²/cm²

A_measured ≈ 10649.4 sq. m

This area represents the field size as measured by the faulty (short) chain.

Step 3: Calculate True Field Area (Correcting for Chain Error)

Finally, correct the measured field area for the known error in the chain length. Use the area correction formula.

True Field Area (A_true) = ( L_act / L_nom )² × A_measured

A_true ≈ ( 19.92 m / 20 m )² × 10649.4 sq. m

A_true ≈ ( 0.996 )² × 10649.4 sq. m

A_true ≈ 0.992016 × 10649.4 sq. m

A_true ≈ 10564.38 sq. m

Final Result

The true area of the survey is approximately 10564.38 sq. m.

Conceptual Explanation & Applications

Core Concepts:

Plan Distortion (Shrinkage/Expansion): Physical map media (especially paper) can change dimensions over time due to environmental factors. This introduces errors when measuring directly from the distorted plan. Correction requires comparing a known original dimension to its current dimension to find a distortion factor.
Area Distortion Correction: Since area is two-dimensional, the correction factor derived from linear distortion must be squared when adjusting area measurements made on the distorted plan. The formula is: Original Area = Measured Distorted Area × (Original Linear Dimension / Current Linear Dimension)².
Map Scale & Area Conversion: The map’s original scale relates distances on the original (undistorted) plan to ground distances. The area scale, derived by squaring the linear scale factor (e.g., (10 m/cm)² = 100 m²/cm²), converts the original plan area to the measured ground area.
Systematic Instrument Error: Measuring instruments like chains may have an actual length different from their nominal length (e.g., a 20m chain being 19.92m long). This systematic error affects all measurements made with that instrument.
Instrument Error Correction (Area): To find the true area from an area measured with a faulty instrument, the measured area must be corrected using the square of the ratio of the actual instrument length to its nominal length: True Area = Measured Area × (Actual Length / Nominal Length)².
Sequential Correction: When multiple sources of error or transformation exist (like plan distortion and instrument error), they must typically be corrected in a logical sequence that reverses the process by which the final measurement was obtained.

Real-World Applications:

Historical Cartography & Research: Accurately extracting quantitative data (distances, areas) from old maps requires accounting for paper shrinkage or expansion.
Land Surveying & Boundary Retracement: Interpreting historical survey plans often involves dealing with potential distortions in the plan medium and uncertainties about the accuracy of original measuring equipment.
Geographic Information Systems (GIS): When digitizing historical maps to create digital data layers, corrections for scale and media distortion are essential for accurate georeferencing and spatial analysis.
Legal & Property Disputes: Resolving boundary disputes may involve analyzing old maps and survey records, necessitating corrections for known or estimated errors in both the plans and original measurements.
Engineering Project Planning: Using existing historical site plans for preliminary design requires assessing their reliability and correcting for potential distortions or inaccuracies before relying on the data.

Why It Works:
The solution follows a logical sequence to reverse the processes that led from the true ground area to the current measurement on the shrunk plan. First, the physical distortion of the map is addressed. We know the current area measured on the shrunk plan (100.2 cm²) and how much a known line has shrunk (10 cm to 9.7 cm). By applying the squared inverse of the linear shrinkage factor ((10/9.7)²), we calculate the area as it would have appeared on the original, undistorted plan (106.494 cm²). Second, this original plan area is converted to the corresponding ground area using the map’s original scale (1 cm = 10 m). The area scale factor (10² = 100 m²/cm²) transforms the original plan area into the ground area *as it was measured by the survey crew* (10649.4 sq. m). Third, this measured ground area is still inaccurate because the tool used (the chain) was faulty (19.92 m instead of 20 m). We apply the instrument area correction factor, which is the square of the ratio of actual to nominal length ((19.92/20)²), to this measured ground area. This final correction yields the true ground area of the survey (10564.38 sq. m), accounting for both the distortion of the map record and the inaccuracy of the original measuring tool. Each step isolates and corrects for a specific source of error in the appropriate order.