At the end of survey of a field a 30 m chain was found to be 10 cm too long. The area of the plan drawn with the measurements taken with this chain is found to be 125 cm2. If the scale of the plan is 1cm = 10 m, what is the true area of the field.

Problem Statement

At the end of surveying a field, a 30 m chain was found to be 10 cm too long. The area of the plan drawn using the measurements taken with this chain was found to be 125 cm². The scale of the plan is 1 cm = 10 m. Assuming the chain was exactly 30 m at the commencement of the work, what is the true area of the field?

Step-by-Step Solution

Key Information

Nominal Length of Chain (L_nom) = 30 m
Actual Length at Start (L_start) = 30.0 m (Error = 0 cm)
Actual Length at End (L_end) = 30 m + 10 cm = 30.1 m (Error = +10 cm)
Plan Area (A_plan) = 125 cm²
Scale = 1 cm : 10 m (Scale Factor S = 10 m/cm)
Assumption: Chain length error increased linearly during the survey.
Goal: Find the True Field Area (A_true).

Step 1: Calculate Measured Field Area from Plan

The area on the ground (field area) represented by the plan area is found by multiplying the plan area by the square of the scale factor.

Measured Field Area (A_measured) = A_plan × (Scale Factor S)²

A_measured = 125 cm² × (10 m / 1 cm)²

A_measured = 125 cm² × (100 m²/cm²)

A_measured = 12,500 sq. m

This is the area calculated based on the measurements taken with the inaccurate chain.

Step 2: Determine Average Chain Length for Correction

Since the chain error changed linearly from 0 cm at the start to +10 cm at the end, we use the average chain length for the correction calculation.

Average Actual Length (L_avg) = (L_start + L_end) / 2

L_avg = (30.0 m + 30.1 m) / 2

L_avg = 30.05 m

Step 3: Calculate True Field Area

Apply the area correction formula using the average actual chain length (L_avg) and the nominal length (L_nom).

A_true = ( L_avg / L_nom )² × A_measured

A_true = ( 30.05 m / 30 m )² × 12,500 sq. m

A_true = ( 1.001667 )² × 12,500 sq. m

A_true ≈ 1.003336 × 12,500 sq. m

A_true ≈ 12541.7 sq. m

Final Result

The true area of the field is approximately 12541.7 sq. m.

Conceptual Explanation

Key Concepts Applied:

Plan Scale Conversion: The area measured on a plan (map) must be converted to the actual ground area using the square of the linear scale factor. This is because area is two-dimensional.
Variable Systematic Error: The problem states the chain’s error changed during the survey (from exact to 10 cm too long). When error changes predictably (assumed linearly here), the average error or average instrument length over the measurement period provides the best basis for correction.
Area Correction Factor: As in the previous problem, the correction for area measurements uses the square of the linear correction factor (Average Actual Length / Nominal Length)². This accounts for the error affecting both dimensions contributing to the area calculation.
Combined Corrections: This problem demonstrates combining two common surveying adjustments: converting scaled plan measurements to field measurements and correcting for instrument (chain) error. The measured field area derived from the plan must then be corrected for the chain inaccuracy.