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Horizontal Curve Long Chord

C=2Rsin⁡Δ2C = 2R\sin\frac{\Delta}{2}

Worked example: 200 m chord at Δ = 100 gon → 141.42 m radius — press Try an example to run it live, then adjust anything.

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See also
Deflection Angle to a Point on a Circular Curve — The same half-angle applied part-way along the curve, which is how the intermediate points get staked.
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UniversityApplied Field Engineering

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Horizontal Curve Long Chord explained

ΔRC

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.

Horizontal Curve Long Chord formula

C=2Rsin⁡Δ2C = 2R\sin\frac{\Delta}{2}
Where
  • CC= Long chord (m)
  • RR= Curve radius (m)
  • Δ\Delta= Deflection angle (°)

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Part of this formula set

Horizontal curve layout

6 formulasSetting out a circular road curve: radius from degree of curve, then tangent, curve length, external distance, middle ordinate and long chord.