Horizontal circular curve

All elements of a simple circular curve (radius, deflection, tangent, length, chord, external, middle ordinate, degree of curve) from any two, with and stations.

A published result changed on 2026-09-19: The horizontal curve, vertical curve, and area by coordinates are now stable, so their results no longer carry the EXPERIMENTAL_TOOL warning; the legacy-unit warning on area moved to the first position. They match FM 5-233 (Army Construction Surveying), the Indiana Design Manual vertical-curve example, and Wikipedia's shoelace example. No numbers changed. Changelog

Computing your own values needs JavaScript and WebAssembly. The worked example below was computed when this page was built, and the documentation under "How we got this" applies.

500ft

The curve has tangent 133.975 ft, length 261.799 ft, and long chord 258.819 ft.

Deflection Δ
Tangent T
Curve length L
Long chord C
External E
Middle ordinate M

Provenance

Computed by: survey.curves.circular-curve 1.0.0, core 0.1.0
Model: T = R·tan(Δ/2), L = R·Δ, C = 2R·sin(Δ/2), E = R(sec(Δ/2) − 1), M = R(1 − cos(Δ/2)); D(arc) = 5,729.578/R ft, sin(D(chord)/2) = 50/R ft; PT station = PC + L along the arc
Accuracy: Exact; two given elements other than R and Δ are solved by bisection to 1e-15 relative
Notes: None
Cites: Ghilani, C. D., and Wolf, P. R., Pearson, Elementary Surveying: An Introduction to Geomatics

Something look off?

Your values

Showing an example. Change anything.

Radius Roptional Like 500 ft · plain numbers mean ft Deflection Δoptional Like 30°00'00" or 30 deg · plain numbers mean degrees Tangent Toptional Distance PI to PC, like 250 ft · plain numbers mean ft PI stationoptional Like 12+34.56

More options 6

Curve length Loptional Along the arc, like 480 ft · plain numbers mean ft Long chord Coptional PC to PT, like 470 ft · plain numbers mean ft External Eoptional PI to the curve midpoint, like 30 ft · plain numbers mean ft Middle ordinate Moptional Chord midpoint to the curve midpoint, like 28 ft · plain numbers mean ft Degree of curve (arc)optional Per 100 ft of arc, like 11.4592 deg · plain numbers mean degrees Degree of curve (chord)optional Per 100 ft chord, like 15 deg (railroads and older highway plans) · plain numbers mean degrees

Run many at once from a CSV

Loading…

How we got thisFormula, worked example, sources, and proof

Model: T = R·tan(Δ/2), L = R·Δ, C = 2R·sin(Δ/2), E = R(sec(Δ/2) − 1), M = R(1 − cos(Δ/2)); D(arc) = 5,729.578/R ft, sin(D(chord)/2) = 50/R ft; PT station = PC + L along the arc

Show your work

Radius
R = the radius the two given elements fix
from radius and delta = 500 ft
Tangent
$T = R × tan(Δ/2)$
500 ft × tan(30° / 2) = 133.975 ft
Curve length
L = R × Δ in radians
500 ft × 0.523599 rad = 261.799 ft

The same steps an agent gets from the MCP server with explain: true.

Accuracy: Exact; two given elements other than R and Δ are solved by bisection to 1e-15 relative

When to use this: Use this to work out a simple horizontal curve from whichever two elements you have: radius and deflection, tangent and degree of curve, chord and external, and so on. It returns every other element with the PC and PT stations, which is what staking and plan checking need.

Limitations: It describes one simple circular curve: no spiral transitions, no compound or reverse curves, and no superelevation or sight distance, which have their own tools and design tables. Both degree-of-curve definitions are given because the arc and chord definitions differ; use the one your agency's standard names.

Worked example: R = 500 ft, Δ = 30°. Source: add-survey-suite scenario: T 133.975, L 261.799, C 258.819, E 17.638, M 17.037 ft.

You enter

Deflection Δ: 30 deg
PI station: 12+34.56
Radius R: 500 ft

You get

Radius R: 500 ft
Deflection Δ: 30°
Tangent T: 133.975 ft
Curve length L: 261.799 ft
Long chord C: 258.819 ft
External E: 17.638 ft
Middle ordinate M: 17.037 ft
Degree of curve (arc): 11.4592°
Degree of curve (chord): 11.4783°
PC station: 11+00.59
PT station: 13+62.38

Review: Not yet independently reviewed by a licensed surveyor.

Last verified: 2026-09-23, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.0, core 0.1.0. See this tool in the verification report.

Changes

2026-09-19, result change: The horizontal curve, vertical curve, and area by coordinates are now stable, so their results no longer carry the EXPERIMENTAL_TOOL warning; the legacy-unit warning on area moved to the first position. They match FM 5-233 (Army Construction Surveying), the Indiana Design Manual vertical-curve example, and Wikipedia's shoelace example. No numbers changed. Changelog
2026-09-19, fixed: The circular curve now solves from any two elements in every case: external with middle ordinate failed (the formulas lost all precision near zero), and a tangent with a middle ordinate, which fit two different curves, could fail or pick one silently. It now returns the flatter curve and names the other with AMBIGUOUS_INPUT. A chord-definition degree of curve (degree_chord) is accepted as an input, as the survey spec requires. Changelog
2026-09-19, fixed: An input so small that it rounds to zero in meters (like 5e-324 ft) is now refused as out of range for every tool, instead of being treated as zero; bounds fuzzing found it through the circular-curve tool. A non-positive curve element is now reported on the field that holds it. Changelog

Checked against: 20 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

Elementary Surveying: An Introduction to Geomatics, Ghilani, C. D., and Wolf, P. R., Pearson, 16th edition. Chapters 10 (traverse computations), 12 (area), 24 (horizontal curves), 25 (vertical curves), 26 (volumes).

Terms

PC: point of curvature: On a horizontal curve, where the curve leaves the back tangent. Source: Elementary Surveying: An Introduction to Geomatics
PI: point of intersection: On a horizontal curve, where the two straight tangents would meet. Source: Elementary Surveying: An Introduction to Geomatics
PT: point of tangency: On a horizontal curve, where the curve joins the forward tangent. Source: Elementary Surveying: An Introduction to Geomatics

PC — point of curvature

On a horizontal curve, where the curve leaves the back tangent.

Source: Elementary Surveying: An Introduction to Geomatics

PT — point of tangency

On a horizontal curve, where the curve joins the forward tangent.