Slope correction for chaining on sloping ground – exact expression using height difference If h is the difference in elevation between the endpoints of a measured sloping line of length l, what is the required slope correction C_s (to be subtracted from l) to obtain the horizontal distance?

Introduction / Context:
On sloping ground, tapes and chains usually follow the slope, yielding a measured sloping distance l. For mapping and coordinate computations, we need the corresponding horizontal distance. The difference between the sloping length and its horizontal projection is the slope correction C_s, which must be subtracted from l. This question asks for the exact expression in terms of the measured sloping length l and the elevation difference h between endpoints.

Given Data / Assumptions:

• Measured sloping length between two points = l.
• Difference in elevation between endpoints = h.
• Ground is straight between endpoints (right triangle model), and refraction/curvature are neglected.

Concept / Approach:

The sloping line l, horizontal projection L_h, and vertical difference h form a right triangle. By Pythagoras, L_h = √(l^2 − h^2). The slope correction C_s is defined as the amount to subtract from l to obtain L_h, so C_s = l − L_h. For small slopes (h ≪ l), a binomial expansion gives the useful approximation C_s ≈ h^2 / (2l), but the exact expression remains l − √(l^2 − h^2). Note that C_s is always positive and subtractive from l to yield the shorter horizontal distance.

Step-by-Step Solution:

Verification / Alternative check:

Numerical trials confirm that for gentle slopes, the approximate formula closely matches the exact value, and that C_s increases with h while remaining < l for any feasible survey slope.

Why Other Options Are Wrong:

Option B omits the factor 1/2, overcorrecting at small slopes.

Option C is an unjustified rule-of-thumb and not dimensionally linked to l.

Option D has the wrong sign (would produce a negative correction).

Common Pitfalls:

Adding the correction instead of subtracting it; using h instead of h^2; confusing exact and approximate formulas without checking slope magnitude.

Final Answer:

C_s = l − √(l^2 − h^2)