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Horizontal Curve External Distance

E=R(sec⁡Δ2−1)E = R\left(\sec\frac{\Delta}{2} - 1\right)
sig figs

Worked example: 20 m external at Δ = 100 gon → 48.284 m radius. Press Try an example to run it live, then adjust anything.

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Where Horizontal Curve External Distance fits

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See also

Horizontal Curve Long Chord: 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.Deflection Angle to a Point on a Circular Curve: Both use curve radius.Minimum Radius of a Horizontal Curve at a Design Speed (AASHTO): Both use curve radius.
2 more relatedRadius from Degree of Curve (Arc Definition): Both use curve radius.Superelevation Rate for a Horizontal Curve: Both use curve radius.
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Horizontal Curve External Distance explained

ΔRE

The external distance answers the right-of-way question: how far does the pavement stay from the corner the two tangents would have made? Measured from the PI along the bisector to the midpoint of the arc, E is the gap that has to clear whatever sits at the corner: a rock face, a building, a wetland boundary. Because the centre, the PI and the curve midpoint are collinear, R + E is the hypotenuse of the same half-angle triangle that produced the tangent length, giving the secant form.

A worked example: a 500 ft curve deflecting 60° has E = 500(sec 30° − 1) = 500(1.1547 − 1) = 77.35 ft. If the survey shows a building face 60 ft inside the PI on the bisector, that curve will not fit and either the radius has to shrink or the alignment has to shift. Watch the behaviour at large Δ: as the deflection approaches 180° the secant blows up and E runs away to infinity, which is the geometry's way of saying a hairpin needs a small radius, not a big one.

Horizontal Curve External Distance formula

E=R(sec⁡Δ2−1)E = R\left(\sec\frac{\Delta}{2} - 1\right)
Where
  • EE= External distance (m)
  • RR= Curve radius (m)
  • Δ\Delta= Deflection angle (°)
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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
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formula.expert. (2026, October 16). Horizontal Curve External Distance Formula & Calculator. https://formula.expert/solvers/circular-curve-external-distance

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