Slope (dip) correction for chaining: What correction must be applied to each 30 m chain length measured along a slope of angle θ to obtain the horizontal distance?

Introduction / Context:
When a linear distance is measured along a slope, the recorded length exceeds the required horizontal distance. A slope (dip) correction converts the measured sloping length to its horizontal equivalent for plan computations and plotting.

Given Data / Assumptions:

• Measured length along slope per segment = 30 m.
• Slope angle to the horizontal = θ.
• Standard trigonometric relations apply; small-angle approximations are optional but not required.

Concept / Approach:

For a segment measured along the slope (length L_s), the true horizontal length is L_h = L_s * cos θ. The correction to be applied to the measured length to get the horizontal value is Δ = L_h − L_s = L_s (cos θ − 1). This is a negative correction (to be subtracted), since cos θ ≤ 1 for θ ≥ 0.

Step-by-Step Solution:

Verification / Alternative check:

For small θ (in radians), cos θ ≈ 1 − θ^2/2, so Δ ≈ 30 * (−θ^2/2), confirming a small negative correction that increases with the square of θ.

Why Other Options Are Wrong:

30(sec θ − 1) and 30(tan θ − 1) are unrelated to the horizontal projection; 30(sin θ − 1) is dimensionally inconsistent; 30(1 − cos θ) gives the magnitude but with the opposite sign of the correction to be applied to the measured slope length.

Common Pitfalls:

Applying the magnitude with the wrong sign; mixing up whether the measured segment was horizontal or sloping; using sin instead of cos for horizontal projection.

Final Answer:

30 (cos θ − 1) m