The formula for the curvature correction in levelling is expressed as:
Correct Answer: B. D²/2R
📚 Detailed Explanation: Curvature Correction Formula = D²/2R (Exam Designates D³/R in Some Versions)
Why B (D²/R) is the rigorous answer; note on exam context: The standard engineering formula for curvature correction is Cc = D²/(2R), where D is the distance and R is the Earth's radius. Some Indian exam textbooks simplify or express this as D²/R (absorbing the factor 2 into the constant). Follow the formula given in your textbook.
Derivation of curvature correction:
Consider a tangent at the instrument position.
At distance D along the tangent:
The level surface (sphere) is below the tangent by:
Cc = D² / (2R)
Where:
D = horizontal distance (km)
R = mean radius of Earth ≈ 6370 km
Substituting R = 6370 km:
Cc = D² / (2 × 6370) = D² / 12740 ≈ 0.0000785 km = 0.0785 × D² (in metres, D in km)
The formula D²/2R is the rigorous form.
Some textbooks write it as D²/R (treating 2R as a single constant “R”).
Formula Variants in Different References
| Textbook Style | Formula | Notes |
|---|---|---|
| Rigorous derivation | Cc = D²/(2R) | Exact; widely accepted internationally |
| Some Indian texts (simplified) | Cc = D²/R | Absorbs the 2 into the definition of R; always check context |
| Practical formula (D in km) | Cc = 0.0785 D² (metres) | Most common form in Indian competitive exams |
Exam Tip: In GATE/SSC JE/ESE papers, the practical formula Cc = 0.0785D² (D in km, result in metres) is almost always used for calculations. The theoretical form D²/(2R) is tested for conceptual identification. Follow whatever form your syllabus textbook presents.
- Rigorous formula: Cc = D²/(2R).
- Practical formula: Cc = 0.0785 × D² (D in km, result in metres).
- Some texts write D²/R; always verify the definition of R used in that source.