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Horizontal curve calculator

Free and built for road and site layout: solve a simple circular curve from any two of its elements, with PC and PT stations from the PI.

  • Runs in your browser
  • Every answer shows its formula

Why this one

Built for drawings, not office PDFs

  • Any two elements

    Solve from radius and delta, degree of curve and delta, or delta with the length, tangent, chord, external, or middle ordinate.

  • Angles the way surveyors write them

    Type delta in degrees, minutes, and seconds or decimal degrees, and read it back both ways.

  • Stationing done right

    PC from the PI and the tangent, and PT along the curve, in feet or metric stations.

The parts of a horizontal curve

A simple horizontal curve joins two straight tangents with a circular arc. The tangents meet at the point of intersection (PI). The curve begins at the point of curvature (PC) and ends at the point of tangency (PT). The deflection between the tangents is the delta angle (Δ), which is also the central angle of the arc. The radius (R) sets how sharp the curve is, and from R and Δ every other element follows.

How to use the horizontal curve calculator

  • Choose the pair of values you know. Radius and delta is the most common; the others cover a curve measured off a plan or set by a constraint.
  • Type the two values. Delta can be decimal degrees or degrees, minutes, and seconds.
  • Add the PI station if you want stationing, written as 12+50.00 in feet or 1+250.000 in meters.
  • Read the results. Each element shows the formula used, and the diagram is drawn at your delta.

The formulas

  • Length of curve: L = R x Δ, with Δ in radians, or πRΔ / 180 with Δ in degrees.
  • Tangent: T = R tan(Δ/2). Long chord: C = 2R sin(Δ/2).
  • External: E = R(sec(Δ/2) - 1). Middle ordinate: M = R(1 - cos(Δ/2)).
  • Degree of curve, arc definition: D = 5729.578 / R, in feet.
  • Stations: PC = PI - T, and PT = PC + L.

The stationing trap

A frequent error on curve tables is computing the PT as the PI station plus the tangent. Stationing runs along the alignment, and once the alignment leaves the tangent at the PC it follows the arc, which is shorter than the two tangents together. So PT = PC + L. For a 500-foot radius with a 30° delta and the PI at 12+50.00, the PC is 11+16.03 and the PT is 13+77.82, while PI + T would give 13+83.97.

Checking a curve on a plan

Curve tables on plans list R, Δ, L, and often T and C. To check one, solve from R and Δ and compare the rest. If the tangent and chord match but the length doesn't, the length was probably measured along the chord. If nothing matches, check whether delta was written in decimal degrees or in degrees and minutes.

Curves in metric

Switch the units to meters for metric work. Every length is then in meters, and stations are written per kilometer, such as 1+250.000. Degree of curve is left out in metric, since it is defined by a 100-foot arc; metric practice states curves by their radius alone.

FAQ

Questions, answered.

Something else? Read the Vellum documentation or talk to the team.

What do I need to solve a horizontal curve?

Any two independent elements, usually the radius and the delta (central) angle. From those, every other element follows: L = R x Δ (in radians), T = R tan(Δ/2), C = 2R sin(Δ/2), E = R(sec(Δ/2) - 1), and M = R(1 - cos(Δ/2)). The calculator also accepts delta with any one length, or the radius with the length, tangent, or chord.

How do I enter the delta angle?

As decimal degrees (45.5) or degrees, minutes, and seconds: 45°30'15", 45 30 15, or 45-30-15 all work. Results show delta both ways.

How are the PC and PT stations calculated?

The PC is the PI station minus the tangent length. The PT is the PC plus the length of the curve, not the PI plus the tangent, because stationing follows the curve once it leaves the tangent. Feet stations are written per 100 feet (12+34.56) and metric stations per kilometer (1+234.567).

What is the degree of curve?

The central angle that a 100-foot length of curve subtends, an older way of stating sharpness used in US highway work. Under this arc definition, D = 5729.578 / R. Railroads have traditionally used a chord definition instead, which gives slightly different values for sharp curves. The calculator uses the arc definition, in feet.

What are the external and middle ordinate?

The external (E) is the distance from the PI to the middle of the curve. The middle ordinate (M) is the distance from the middle of the long chord to the middle of the curve. Both come up when checking clearances and sight lines on the inside of a curve.

Keep going

Related free tools

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Put the curve on the drawing.

In Vellum, draw arcs at a typed radius and sweep, fillet and offset them, and measure them at the drawing's scale.

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