Angular form of slope correction: If θ is the ground slope angle and l is the measured sloping distance between two points, what is the correction to reduce l to horizontal length?

Introduction / Context:

Slope measurements must be reduced to horizontal to compute plan distances and areas correctly. When the ground slope angle θ is known, the correction can be expressed in a compact trigonometric form useful for quick computations and field tables.

Given Data / Assumptions:

• Slope distance measured along the ground = l.
• Ground makes an angle θ with the horizontal.
• We need the correction to subtract from l to obtain horizontal distance.

Concept / Approach:

The horizontal distance is D = l cos θ. The required correction c = l − D = l(1 − cos θ). Using the identity 1 − cos θ = 2 sin²(θ/2), we get c = 2l sin²(θ/2), which is convenient for small-angle work and matches series forms used in approximations.

Step-by-Step Solution:

Verification / Alternative check:

For small θ, sin(θ/2) ≈ θ/2 (in radians), giving c ≈ 2l (θ²/4) = l θ²/2, consistent with the series c ≈ h²/(2l) when h ≈ l sin θ and θ is small.

Why Other Options Are Wrong:

• 2l cos²(θ/2) and others do not reduce to l(1 − cos θ).
• l sin θ gives vertical difference, not the correction.

Common Pitfalls:

• Subtracting the wrong component (vertical instead of horizontal).

Final Answer:

2l sin²(θ/2)