Free Surveying Tool
Horizontal Curve Calculator
Solve all curve elements from any two known values. Works with arc or chord definition. DMS and decimal degree input.
Fill in any two values — the rest calculate automatically.
Intersection angle at the PI
R = 5729.578 / D
D = 5729.578 / R (arc def.)
T = R · tan(Δ/2)
L = π · R · Δ / 180
LC = 2R · sin(Δ/2)
E = R · (sec(Δ/2) − 1)
M = R · (1 − cos(Δ/2))
Curve Element Reference
All values use the arc definition (D° = 5729.578 / R), standard for U.S. land surveying.
The deflection angle at the PI — equal to the angle subtended by the arc at the center of the circle. Also called the intersection angle.
Horizontal distance from the center of the circle to any point on the curve. Larger R means a gentler curve.
The central angle subtended by a 100-foot arc. Arc definition: D = 5729.578 / R. Common in U.S. highway and land surveying.
Distance from the PI (Point of Intersection) to the PC or PT along the tangent line. T = R · tan(Δ/2).
Total length along the curve from PC to PT. L = π · R · Δ / 180. This is what you chain along the curve.
Straight-line distance from PC (Point of Curvature) to PT (Point of Tangency). LC = 2R · sin(Δ/2).
Distance from the PI to the midpoint of the curve, measured along the bisector of the intersection angle. E = R · (sec(Δ/2) − 1).
Distance from the midpoint of the long chord to the midpoint of the curve. M = R · (1 − cos(Δ/2)).
Horizontal Curve Formulas
All formulas assume arc definition and angles in decimal degrees unless noted.
Solving from Two Knowns
R + Δ — The most common field scenario. You know the radius from the plat and the deflection angle from your traverse. All other elements follow directly.
D + Δ — Highway plans often express curves as Degree of Curve rather than radius. Convert first: R = 5729.578 / D.
T + L (iterative) — When you can measure both the tangent distance and the arc length in the field but have no other data, this pair requires iteration. The calculator uses bisection to converge on Δ, then derives R.
Δ + E or Δ + M — Useful when you can observe the intersection angle and measure the external or middle ordinate directly in the field.
R + LC (or Δ + LC) — Long chord can be measured with a total station or tape between PC and PT even when the curve itself is obstructed.
Arc vs. Chord Definition
This calculator uses the arc definition, where D° is the central angle subtended by a 100-foot arc. This is standard in U.S. land surveying and most state DOT work. Railroad surveying historically used the chord definition (D° subtended by a 100-foot chord), which produces slightly different R values for the same D° — the difference is negligible for flat curves but grows for sharp ones.
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