What this calculator does
Every bend in a road or track is a circular curve with five numbers that define it. This calculator returns those numbers — tangent length, external distance, middle ordinate, curve length and long chord — and, if you enter superelevation and side friction, the design speed the curve will hold.
How to use it
- Enter the deflection (intersection) angle and the chosen radius.
- Enter the design superelevation as a percentage and the side-friction coefficient.
- The tangent and external distance are what you stake on site; the middle ordinate is what you need to set out the curve itself.
- The speed check uses V² = gR(e + f); the result should not exceed the design speed of the road.
Governing formula
T = R·tan(Δ/2) ; E = R(sec(Δ/2) − 1) ; M = R(1 − cos(Δ/2)) Where:
-
Δ= Deflection angle [°] -
R= Curve radius [m]
Derived Equations:
L = πRΔ/180 C = 2R·sin(Δ/2) V = √(gR(e + f)) Worked example — A 35° bend on a 120 m radius
The figures below were produced by running this page's own engine with the example inputs (Deflection angle (Δ) (°), Curve radius (m), Superelevation (e) (%), Side friction (f)). If the calculator returns something different for the same numbers, that is a bug, not a rounding difference.
| Tangent length (T) | 37.84 m |
|---|---|
| Curve length (L) | 73.3 m |
| External distance (E) | 5.82 m |
| Middle ordinate (M) | 5.55 m |
| Long chord (C) | 72.17 m |
| Comfortable speed | 52.4 km/h |
| Assessment | PI to PC 37.8 m, curve length 73.3 m, middle ordinate 5.6 m — comfortable design speed ≈ 52 km/h. |
- T = 120 × tan(17.5°) ≈ 37.8 m, so set back that distance from the point of intersection on each tangent.
- The external distance E ≈ 6.2 m is the offset between the PI and the midpoint of the curve.
- With 6 % superelevation and f = 0.12 the curve holds roughly 45 km/h.
Assumptions & limitations
Engineering calculation. Circular-curve geometry from the intersection angle and radius, plus a speed check from superelevation and side friction. No spiral transitions, cant or sight-distance analysis.
- Circular curves only — no spiral (transition) curves, so the result does not match highway-standard geometry that requires them.
- No sight-distance, crest or sag vertical-curve interaction, or overtaking sight distance check.
- The speed check assumes a constant superelevation across the curve.
This tool supports professional engineering work; it does not replace a sealed design. Confirm every result against the applicable standard and a licensed engineer before it is built.
References & standards
- AASHTO Green Book — geometric design of roads and highways.
- BS EN 13223 — road geometry, horizontal curves.
- Highway Capacity Manual — side friction and superelevation relationships.