Horizontal Curve Calculator (Circular Curve Elements & Stakeout)
Calculate the elements of a circular horizontal curve — tangent length, curve length, long chord, external and middle ordinate, degree of curve, PC and PT stations and a deflection-angle stakeout table.
How it works
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)] · PC = PI − T, PT = PC + L · degree of curve (arc) D = 5,729.58/R for a 100-ft arc or 1,145.92/R for a 20 m arc · deflection to a point at arc length l from the PC = l/(2R) radians, chord = 2R·sin(deflection).
A simple horizontal curve is a circular arc joining two straight tangents that meet at the point of intersection (PI) at a deflection angle Δ. Its elements follow from the geometry of the circle: the tangent length from PC to PI, the arc length, the long chord, the external distance from the PI to the arc and the middle ordinate from the chord to the arc. Stations along the curve are measured along the arc, so the PT station is PC + L, not PI + T. For stakeout, a total station at the PC turns the deflection angle (half the central angle) to each station and measures the chord, as described in standard route-surveying texts and the AASHTO Green Book.
Worked example
A curve of radius 300 m with Δ = 40° and the PI at station 1+250.00: T = 300 × tan 20° = 109.19 m, L = π × 300 × 40/180 = 209.44 m, LC = 2 × 300 × sin 20° = 205.21 m, E = 19.253 m and M = 18.092 m. The PC is at 1+140.81 and the PT at 1+140.81 + 209.44 = 1+350.25. The first full station, 1+160, is 19.19 m along the arc, so its deflection angle is 19.19/(2 × 300) rad = 1°49′57″.
Sources: Horizontal curve — circular curve elements and stationing · FHWA — Flexibility in Highway Design, Chapter 6: Horizontal Alignment
Frequently asked questions
Why isn’t the PT station equal to the PI station plus the tangent?
Stationing follows the road centreline, which runs along the arc. The arc (L) is shorter than the two tangents (2T), so PT = PC + L. Using PI + T would give the station along the original tangent, not along the road.
What is the degree of curve?
An older US way to express sharpness: the central angle subtended by a 100-ft arc (arc definition, D = 5,729.58/R) or by a 100-ft chord (chord definition, used by railways). Metric practice simply quotes the radius, sometimes with a 20 m arc degree.
How do I stake out a curve with deflection angles?
Set up at the PC, sight the PI with zero on the instrument, turn the deflection angle for each station and measure the chord from the PC (or the chord between consecutive stakes). The deflection to the PT should equal Δ/2 — a good check.
What radius should a road curve have?
It depends on design speed, superelevation and side friction: AASHTO’s minimum radius is V²/[127(e + f)] in metric. At 80 km/h with 6% superelevation (f = 0.14) that is about 250 m; higher speeds need much flatter curves.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.