Horizontal Curve Long Chord

C = 2R\sin\frac{\Delta}{2}

sig figs

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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Where Horizontal Curve Long Chord fits

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RRadius from Degree of Curve (Arc Definition) · Minimum Radius of a Horizontal Curve at a Design Speed (AASHTO)

In this toolkit

Horizontal curve layout6 formulas

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.Horizontal Curve External Distance: Both use curve radius and deflection angle.Horizontal Curve Middle Ordinate: Both use curve radius and deflection angle.Horizontal Curve Tangent Length: Both use curve radius and deflection angle.Horizontal Curve Length from Degree of Curve: Both use deflection angle.Minimum Radius of a Horizontal Curve at a Design Speed (AASHTO): Both use curve radius.

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Horizontal Curve Long Chord explained

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 = 2R\sin\frac{\Delta}{2}

Where

$C$ = Long chord (m)
$R$ = Curve radius (m)
$\Delta$ = Deflection angle (°)

In this toolkit

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.

Revised 16 October 2026

Cite this page

formula.expert. (2026, October 16). Horizontal Curve Long Chord Formula & Calculator. https://formula.expert/solvers/circular-curve-long-chord

https://formula.expert/solvers/circular-curve-long-chord