Horizontal circular curve calculator: tangent, length, stations and stakeout table
From the central angle Δ (the deflection angle at the PI), the radius and the PI station, you get the tangent length, the length of curve, the PC and PT stations and a stakeout table station by station. Enter the PI coordinates and the azimuth to get the stakeout by coordinates too.
Coordinates (optional): PI and azimuth of the back tangent
Stations (chainage) in km+m format: 1+245.300 = 1,245.300 m from the start of the alignment; plain meters (1245.3) work too. The angle accepts decimal degrees or degrees/minutes/seconds.
Result
Enter the central angle, the radius and the PI station.
Simple circular curve formulas
- Tangent length: T = R · tan(Δ/2) · Length of curve: L = π · R · Δ / 180
- External distance (PI to the midpoint of the curve): E = R · (sec(Δ/2) − 1) · Middle ordinate: M = R · (1 − cos(Δ/2))
- Long chord PC–PT: LC = 2 · R · sin(Δ/2) · Degree of curve, chord definition (chord c): D = 2 · arcsin(c / 2R)
- Stations: PC = PI − T; PT = PC + L (the PT comes from the length of curve, not from the tangent)
- Deflection angle of a point at an arc distance s from the PC, measured from the tangent: δ = s / (2R) (in radians); the chord to it is 2 · R · sin(δ)
FAQ
How do I compute the PC and PT stations?
PC station = PI station − tangent length (T). PT station = PC station + length of curve (L). Careful: the PT is not PI + T, because the route along the curve is shorter than along the two tangents.
What is the degree of curve?
This tool uses the chord definition: the central angle subtended by a standard chord of 20 m (10 m when the station interval is 10 m): D = 2 · arcsin(10/R) for the 20 m chord. The larger the radius, the smaller the degree; the deflection per meter is D/(2·20). It is not the same number as the US arc definition (central angle of a 100 ft arc, D = 5729.58/R with R in feet).
How is the curve staked out by deflection angles?
Set up the theodolite or total station on the PC and zero it on the PI (along the tangent). For each station, turn the cumulative deflection angle from the table and measure the chord from the PC (or the chord from the previous station). The table on this page gives both.
Can I stake out by coordinates?
Yes: open “Coordinates” and enter the E and N of the PI and the azimuth of the back tangent (from the PC to the PI). The table then lists the E and N of every station, ready for a total station or GNSS RTK.