Free Surveying Tool
Horizontal Curve Calculator
Solve all curve elements from any two known values. Arc or chord definition, PC and PT stationing, DMS and decimal degree input.
Degree of Curve definition
Arc — D is subtended by a 100 ft arc. U.S. highway and land surveying standard.
D = 5729.578 / R
Fill in any two values — the rest calculate automatically.
Intersection angle at the PI
R = 5729.578 / D
D = 5729.578 / R
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
Every element below is derived from the radius and the central angle, so it is the same under either definition of degree of curve.
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 100 feet of curve — measured along the arc under the arc definition, or along a chord under the chord definition.
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)).
Degree of Curve to Radius Chart
Radius in feet for common degrees of curve, under both definitions. The two agree closely on flat curves and separate as the curve sharpens.
Show the chartHide the chart · 22 rows, D = 0° 30' to 30°
| Degree of Curve | Arc Radius (ft) | Chord Radius (ft) | Diff (ft) |
|---|---|---|---|
| 0° 30' | 11459.16 | 11459.19 | 0.04 |
| 1° | 5729.58 | 5729.65 | 0.07 |
| 1° 30' | 3819.72 | 3819.83 | 0.11 |
| 2° | 2864.79 | 2864.93 | 0.15 |
| 2° 30' | 2291.83 | 2292.01 | 0.18 |
| 3° | 1909.86 | 1910.08 | 0.22 |
| 3° 30' | 1637.02 | 1637.28 | 0.25 |
| 4° | 1432.39 | 1432.69 | 0.29 |
| 4° 30' | 1273.24 | 1273.57 | 0.33 |
| 5° | 1145.92 | 1146.28 | 0.36 |
| 6° | 954.93 | 955.37 | 0.44 |
| 7° | 818.51 | 819.02 | 0.51 |
| 8° | 716.20 | 716.78 | 0.58 |
| 9° | 636.62 | 637.27 | 0.65 |
| 10° | 572.96 | 573.69 | 0.73 |
| 12° | 477.46 | 478.34 | 0.87 |
| 14° | 409.26 | 410.28 | 1.02 |
| 16° | 358.10 | 359.26 | 1.17 |
| 18° | 318.31 | 319.62 | 1.31 |
| 20° | 286.48 | 287.94 | 1.46 |
| 25° | 229.18 | 231.01 | 1.83 |
| 30° | 190.99 | 193.19 | 2.20 |
Arc radius is 5729.578 / D; chord radius is 50 / sin(D/2). For values not listed here, use the calculator above — enter D and any second element to solve the full curve.
Horizontal Curve Formulas
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
The arc definitiontreats D° as the central angle subtended by a 100-foot arc. It is the standard in U.S. land surveying and most state DOT work, and it is this calculator's default.
The chord definition treats D° as the central angle subtended by a 100-foot chord. It is the convention in railroad surveying, where track was historically laid out by chaining a series of 100-foot chords. Use the toggle at the top of the calculator to switch.
For the same D°, the chord definition always produces a slightly larger radius. The gap is negligible on flat curves — about 0.07 ft at D = 1° — but grows as curves sharpen, reaching roughly 2.2 ft at D = 30°, as the chart above shows.
There is a second consequence worth knowing. Under the chord definition, stationing is chained along 100-foot chords rather than along the true arc, so the stationed length of a curve runs slightly short of its arc length. On a D = 10°, Δ = 60° curve the chord chain measures 600.00 ft against a true arc of 600.76 ft. The calculator reports both and uses the chord chain for PT stationing when chord mode is active.
Curve Stationing
Enter the PI station in the calculator and it returns the PC and PT stations. The PC is found by subtracting the tangent distance from the PI station. The PT is then reached by chaining forward along the curve:
PC = PI − T
PT = PC + L
The common mistake is computing the PT as PI + T. That lands at the wrong station, because the route from PC to PT follows the curve rather than the two tangent legs. The difference between them is exactly 2T − L.
Station entry accepts either format: 12+34.56 or 1234.56.
Not sure which curve type your job calls for? See Horizontal, Vertical, and Spiral Curves Explained.
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