Q1. Calculate the area of a field when the initial reading = 9.918 and final reading = 4.254 were recorded by a planimeter. Multiplying constant M = 100 cm², instrument constant C = 23.521, and N = −1 (zero mark passed the fixed index once in reverse direction).
📚 Detailed Explanation: Planimeter Area Calculation — A = 785.7 cm²
A planimeter is a minor surveying instrument used to measure the area of any irregular closed figure drawn on paper — a map plot, a cross-section, or a catchment boundary. The tracer arm is guided around the boundary of the figure and the roller wheel records displacement proportional to the area swept. The difference between initial and final readings is then substituted into the planimeter formula to find area.
M = multiplying constant of the planimeter
F = final reading (after tracing the figure boundary)
I = initial reading (before tracing)
N = number of times the zero mark of the graduated disc passes the fixed index
Use −10N when the zero mark crosses in the reverse direction
C = instrument constant (zero circle area; included only when the anchor point is inside the figure)
Step-by-Step Calculation
M = 100 cm², F = 4.254, I = 9.918, N = -1, C = 23.521
A = M × [F – I ± 10N + C]
A = 100 × [4.254 – 9.918 – 10(1) + 23.521] ↠N = -1, so use -10
A = 100 × [4.254 – 9.918 – 10 + 23.521]
A = 100 × [-5.664 – 10 + 23.521]
A = 100 × [7.857]
A = 785.7 cm²
Why the Other Options Are Wrong
A (895.7 cm²): Results from a sign error in the N correction — using +10 instead of −10 for N = −1. The reverse crossing always means a subtraction, not an addition.
B (986.5 cm²): Likely from misreading the instrument constant C or applying the formula with wrong sign conventions. No standard substitution into the correct formula produces this value.
D (655.7 cm²): Could result from misremembering C as 13.521 instead of 23.521: 100 × [4.254 − 9.918 − 10 + 13.521] = 100 × 6.557 ≈ 655.7. Always read the given C value carefully.
Key Concepts for Students
- When to include C: The instrument constant C is added only when the anchor point is inside the figure being traced. If the anchor is outside (the standard setup for most figures), omit C entirely. This distinction separates Q1 and Q2 in a typical exam.
- F < I is normal: The disc can wrap around during tracing, so the final reading can be numerically smaller than the initial reading. This does not indicate an error in procedure — the N term accounts for this wrap-around.
- Field relevance: Planimeters are used in practice to compute cross-sectional areas from plotted ground profiles, drainage catchment areas from topographic maps, and irregular land areas from survey plans — situations where geometric formulas cannot be directly applied.