Simple circular curve – tangent length relation For a simple circular curve of radius R deflecting through a central angle θ (in degrees), what is the formula for the tangent length?

Introduction / Context:
Tangent length is a fundamental quantity in setting out a simple circular curve in highways and railways. It is the distance from the point of intersection (P.I.) to a tangent point and is required for staking and land acquisition.

Given Data / Assumptions:

• Simple circular curve (single radius).
• Curve radius R.
• Central angle of the curve θ.
• Standard geometry with tangents intersecting at the P.I.

Concept / Approach:
In the right triangle formed by the P.I., center of the curve, and a tangent point, the half-angle θ/2 subtends the tangent from the P.I. to the tangent point. Using basic trigonometry, tangent length T equals R tan(θ/2).

Step-by-Step Solution:
Consider half the curve: angle at the center = θ/2.Opposite side = T; adjacent side = R.tan(θ/2) = T / R → T = R tan(θ/2).

Verification / Alternative check:
For small θ, tan(θ/2) ≈ θ/2 (in radians), so T ≈ R * θ/2, consistent with arc approximations.

Why Other Options Are Wrong:

• R tan θ and R sin θ: use the full angle θ, not the required θ/2 geometry.
• R sin θ/2: gives mid-ordinate related lengths, not the tangent length.
• R cot θ/2: corresponds to external distance relationships, not T.

Common Pitfalls:
Mixing degrees and radians; confusing tangent length with long chord or external distance.

Final Answer:
R tan θ/2