Horizontal Curve Elements Calculator

Work out horizontal curve elements instantly with clear inputs, formula shown and shareable results.

About Horizontal Curve Elements Calculator

All the elements of a simple circular curve follow from the radius and the deflection angle. Tangent length is R tan(Δ/2), curve length is R times Δ in radians, external distance from the intersection point to the curve is R(sec(Δ/2) - 1), and the middle ordinate from the long chord to the curve is R(1 - cos(Δ/2)).

Formula

Curve elements

T = R tan(Δ/2); L = R Δ in radians; E = R(sec(Δ/2) - 1); M = R(1 - cos(Δ/2))

Worked example

With the default values below, Horizontal Curve Elements Calculator returns Length of the curve of 209.4395. Change any input above and every figure here is recalculated in your browser.

Inputs used

Curve radiusminimum 1 m: 300 m
Deflection angleaccepts 0.1–179 °: 40 °

Results

Length of the curve: 209.4395 m
Tangent length: 109.1911 m
External distance: 19.2533 m
Middle ordinate: 18.0922 m

Step by step

1Values used
2Curve elements
3Length of the curve= 209.4395 m
4Tangent length= 109.1911 m
5External distance= 19.2533 m
6Middle ordinate= 18.0922 m

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
Preliminary estimate only. Horizontal alignment must satisfy the design speed, superelevation and sight distance requirements of the applicable standard.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Preliminary estimate only. Horizontal alignment must satisfy the design speed, superelevation and sight distance requirements of the applicable standard.

Frequently Asked Questions

How is the Horizontal Curve Elements figure worked out?

T = R tan(Δ/2); L = R Δ in radians; E = R(sec(Δ/2) - 1); M = R(1 - cos(Δ/2)). All the elements of a simple circular curve follow from the radius and the deflection angle.

Which element is used for setting out?

Tangent length locates the curve start and end from the intersection point. Deflection angles from the tangent, or coordinates from a total station, then set out points along the curve itself.

What values do I need to enter?

This calculator takes 2 inputs: Curve radius, Deflection angle. The pre-filled defaults are a realistic worked example — replace them with your own site or project figures.