Free calculator

Horizontal curve solver

Any two elements in, every element out — plus the deflection table you actually stake from.

Everything is computed in your browser. Nothing you type is uploaded, stored, or logged.

Fill any two of the seven. Every other element follows, because a simple circular curve has exactly two degrees of freedom. Clear a field to hand control to a different pair.

Radius (R)Centreline radius

Central angle (Δ)Decimal degrees or 16-30-00

Tangent distance (T)PC to PI, and PI to PT

Arc length (L)Along the curve, PC to PT

Long chord (C)Straight line PC to PT

Middle ordinate (M)Curve to the mid-point of the long chord

External distance (E)PI to the mid-point of the curve

Lengths carry whatever unit you enter — the solver never assumes one. Degree of curve and the staking table below assume a 100-unit station, which is the US customary convention.

PI stationWritten 100+00.00, or as a plain distance

Radius (R): 1,100.00
Central angle (Δ): 16°30′00″
Tangent (T): 159.49
Arc length (L): 316.78
Long chord (C): 315.68
Middle ordinate (M): 11.38
External (E): 11.50
Degree of curve, arc: 5°12′31″
Degree of curve, chord: 5°12′38″

PC station: 98+40.51
PI station: 100+00.00
PT station: 101+57.28
Chainage lost (2T − L): 2.21
Total deflection at PT (Δ/2): 8°15′00″
Staked points: 4

Deflection angles from the PC, back tangent as the zero, at full stations
Station	Arc from PC	Deflection	Chord from PC	Chord from previous
99+00.00	59.49	1°32′58″	59.49	59.49
100+00.00	159.49	4°09′13″	159.35	99.97
101+00.00	259.49	6°45′29″	258.89	99.97
101+57.28	316.78	8°15′00″	315.68	57.28

The PC is not listed because its deflection is zero by definition. The last row is the PT, and its deflection is 8°15′00″ — half the central angle. If the instrument does not read that when the PT is turned, the setup or the arc length is wrong, and nothing downstream will fix it.

A simple horizontal curve is a single circular arc joining two straight tangents. Seven numbers describe it — radius, central angle, tangent distance, arc length, long chord, middle ordinate and external distance — and they are not seven independent facts. Fix any two and the other five are already decided. That is the whole reason this calculator has one panel of seven fields instead of a menu of solving modes.

Why two values are enough

A circle in a plane has one shape parameter, the radius. An arc of that circle has one more, the angle it turns through. Everything else is a consequence. Write each element as the radius multiplied by a function of the central angle alone and the structure becomes obvious.

T = R · tan(Δ/2)
L = R · Δ            (Δ in radians)
C = 2R · sin(Δ/2)
M = R · (1 − cos(Δ/2))
E = R · (sec(Δ/2) − 1)

Every element is R times a function of Δ. Two knowns therefore fix R and Δ, and R and Δ fix the rest.

Give the solver Δ and any one length and R falls out by division. Give it R and any one length and Δ comes from inverting one of those shape functions. Give it two lengths and neither R nor Δ directly — say T and E — and their ratio still depends on Δ alone, so Δ is recovered from the ratio and R from either length afterwards. The calculator handles all three cases without asking which one you are in.

What each element is on the ground

R, the radius: center of the circle to the alignment. On a design plan it is the number chosen first, from the design speed and the superelevation.
Δ, the central angle: the angle at the center subtended by the curve. It equals the deflection between the two tangents at the PI, which is why it is measured in the field rather than computed.
T, the tangent distance: PC to PI, and PI to PT. The two are equal on a simple curve, which is the property that makes a curve stakeable from the PI.
L, the arc length: PC to PT along the curve. This is what stationing runs along — not the chord, and not the tangents.
C, the long chord: the straight line from PC to PT. Always shorter than L, and the gap grows quickly as Δ increases.
M, the middle ordinate: mid-point of the long chord to the curve. This answers whether a fence line or a building corner clears the arc.
E, the external distance: PI to the mid-point of the curve. This answers whether the curve clears an obstruction sitting at the apex of the tangents.

Degree of curve, and the two definitions of it

US highway plans have historically labelled curves by degree of curve rather than radius, because a degree of curve can be laid off with a transit directly. There are two incompatible definitions of it, and a plan that does not say which one it uses is ambiguous.

arc definition     Dₐ = 100 · 180 / (π R) = 5729.578 / R
chord definition   D𝑐 = 2 · arcsin(50 / R)

Dₐ is the central angle subtended by a 100 ft arc; D𝑐 the central angle subtended by a 100 ft chord.

The arc definition is the one highway agencies use, because stationing runs along the arc and a curve of Dₐ = 2° then contains exactly fifty stations per 100° of central angle. The chord definition is the railroad convention, and it survives in railroad practice because a track curve is laid out as a series of 100 ft chords between spiked points, so the chord is the thing physically measured. Older right-of-way records and some utility alignments follow the railroad usage.

The two definitions agree on flat curves and part company on sharp ones.
Radius (ft)	Dₐ, arc	D𝑐, chord	Difference
5,729.58	1.0000°	1.0000°	0.001%
1,000	5.7296°	5.7320°	0.04%
500	11.4592°	11.4783°	0.17%
200	28.6479°	28.9550°	1.07%
100	57.2958°	60.0000°	4.72%

Stationing the curve, and the trap in it

The PC is found by backing up the tangent distance from the PI, which is straightforward. The PT is not found by running the tangent distance forward from the PI, and that is the single most common error in curve computation.

PC = PI − T
PT = PC + L          ← along the alignment
PT ≠ PI + T

Stationing measures distance along the alignment, and the alignment follows the arc.

Staking by deflection angles

The field method for a circular curve is the inscribed angle theorem. Set the instrument on the PC, sight the PI along the back tangent, zero the plate, and turn a deflection angle for each point to be set. The deflection to a point lying an arc distance l from the PC is half the central angle that arc subtends.

δ = l / (2R)   radians
chord from PC = 2R · sin(δ)

In minutes of arc, δ = 1718.873 · l / R — the form the older field tables are printed in.

Two consequences make this practical. First, deflections are proportional to arc distance, so a table of them is a table of one multiplication. Second, the distance actually taped or shot is the chord, not the arc, and on any curve worth staking those differ. The calculator gives both the chord from the PC — for a total station working from a single setup — and the chord from the previous point, for a chain-and-transit progression down the curve.

The check at the PT

Because the deflection to any point is half its subtended angle, the deflection to the PT is exactly half the central angle. That identity is the field check, and it is free.

If the accumulated deflection at the PT is not Δ/2, either the instrument was not zeroed on the back tangent or the arc length in the table is not the arc length of the curve.
If the deflections check but the chords do not close on the PT, the radius used to compute the chords does not match the one used for the deflections.
If a single intermediate point is off and the rest are good, the sub-chord to it was taped from the wrong previous point — the chord-from-previous column and the chord-from-PC column are not interchangeable.
Setting the PT independently from the PI, along the forward tangent at distance T, is a second and separate determination of the same point. Two methods landing within a hundredth is a check; one method is only a computation.

Long curves staked entirely from the PC accumulate the pointing error of one setup over the whole arc. Standard practice is to move up — occupy a staked point, backsight the PC with the plate set to the deflection already turned, and carry on — which converts one long lever arm into two short ones.

Questions

Which two elements should I enter?

Whichever two you actually have. R and Δ is the usual pair off a design plan; T and Δ is what a deed or a right-of-way map commonly gives; R and L appears in design reports. Pairs that do not include R or Δ work too — T and E, or C and M — because their ratio determines Δ on its own.

Why does the PT station not equal the PI station plus T?

Because stationing follows the alignment, and the alignment follows the arc rather than the two tangents. PT = PC + L. The quantity 2T − L is chainage lost at the curve, and the calculator reports it so you can carry it forward deliberately rather than discover it later.

Arc definition or chord definition for degree of curve?

Highway work uses the arc definition, D = 5729.578/R. Railroad work uses the chord definition, D = 2·arcsin(50/R). They agree closely on flat curves and diverge as the radius shrinks — about 1% apart at R = 200 ft. If a record gives a degree of curve without saying which definition, the era and the type of work are the best clue, and recomputing the recorded tangent distance both ways will usually settle it.

Should I tape the arc distance or the chord distance?

The chord, always — a tape or an EDM measures a straight line. The arc is the stationing distance and the chord is the measured distance, and the two columns in the staking table exist to keep that distinction visible. On a sharp curve at a 100 ft interval the difference is easily a few hundredths.

Does this handle spirals or compound curves?

No. This solves a simple curve — one arc of one radius between two tangents. A compound curve is two simple curves sharing a point of compound curvature and can be solved as two runs of this tool. A spiral transition needs the clothoid parameters and is a different computation.

Is anything I type sent anywhere?

No. The computation runs entirely in your browser. Nothing is uploaded, logged or stored, which is also why the page keeps working once it has loaded and the signal drops.

Sources

Washington State DOT, Highway Surveying Manual M 22-97, Chapter 11 — published state manual; the simple-curve element formulas, the deflection-angle staking procedure and the worked example the default values reproduce
Montana DOT, Survey Manual, Appendix C — published state manual; curve element formulas and the arc versus chord degree-of-curve definitions

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