Horizontal Curve Long Chord

The long chord is what a total station actually measures when it shoots from the PC to the PT: a straight line through the air, not the arc the pavement follows. It is always shorter than the curve length, and the gap grows quickly with deflection — for Δ = 60° the chord is 4.7 % shorter than the arc, for Δ = 120° it is 17 % shorter. Confusing the two is a classic quantity error, because pavement, guardrail and curb are all paid along the arc while the chord is what a tape across the opening gives.

A worked example: a 500 ft radius curve deflecting 60° has C = 2 × 500 × sin 30° = 500 ft exactly, since a 60° chord equals the radius — the same fact that makes a regular hexagon inscribe in a circle with side length R. The arc, by contrast, is RΔ = 500 × 1.0472 = 523.6 ft. Solving for Δ uses arcsine, which returns the minor arc; a curve that wraps more than halfway round its circle needs the supplement.