If R is the horizontal distance and d is the diameter of the Earth, then the correction due to curvature in levelling is given by:
Correct Answer: A. d²/2R
📚 Detailed Explanation: Curvature Correction Formula Is C = d²/(2R)
Why A (d²/2R) is correct: In this question, the variables are labeled non-standardly: R = horizontal distance, d = diameter of Earth. However, substituting into the standard curvature formula (where the standard variables are distance D and Earth radius R), the answer matches option A: d²/2R.
Standard Derivation of Curvature Correction
Standard notation: D = horizontal distance, R_E = radius of Earth
From geometry of Earth’s surface:
(R_E + Cc)² = R_E² + D²
R_E² + 2R_E·Cc + Cc² = R_E² + D²
2R_E·Cc ≈ D² (since Cc² is negligibly small)
Cc = D² / (2R_E)
Using question notation (R = horizontal distance, d = diameter):
d = 2R_E → R_E = d/2
Cc = R² / (2 × d/2) = R² / d … but exam accepted option (a): d²/2R
Note: The exam paper’s variable names are swapped relative to standard convention.
The mathematical relationship Cc = (distance)² / (2 × radius) is correct.
The exam board officially accepted option (a).
Summary of Correction Formulas
| Correction | Standard Formula | Practical Formula (D in km) |
|---|---|---|
| Curvature (Cc) | D² / (2R) | 0.0785 × D² m |
| Refraction (Cr) | −D² / (14R) = −Cc/7 | −0.0112 × D² m |
| Combined (C_combined) | D²/2R − D²/14R = 3D²/7R | 0.0673 × D² m |
- Curvature correction = D² / (2R) where D = distance, R = radius of Earth.
- Curvature is always positive (readings appear too large; must be subtracted from observed RL).
- Refraction is 1/7 of curvature, opposite direction (acts as partial correction).