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Banking Angle Calculator (Curve Superelevation)

Calculate the ideal frictionless banking angle for a curve at a given speed and radius, tan(θ) = v²/(r·g), the physics behind banked roads, race tracks, and railways.

Ideal (frictionless) banking angle
17.67°
Superelevation rate (rise per 100 of run)
31.86%

How it works

When a vehicle (or a train, or a cyclist on a velodrome track) rounds a curve, something has to supply the centripetal force mv²/r that bends its path. A banked curve tilts the road surface so the normal force itself has a horizontal component pointed toward the curve's center; at exactly one angle, that horizontal component alone supplies the full centripetal force with zero sideways friction needed from the tires at all. Dividing the required horizontal force by the required vertical force (which must still support the vehicle's weight) cancels mass entirely and leaves tan(θ) = v²/(r·g) — the 'ideal' or 'frictionless' banking angle.

That same ratio, expressed as a percentage instead of an angle, is the 'superelevation rate' road and rail engineers actually specify on drawings — e.g. a curve banked so tan(θ) = 0.06 is described as '6% superelevation,' a rise of 6 units for every 100 units across the roadway.

Because v²/(r·g) is a ratio of consistent speed-squared to consistent length-times-acceleration units, it comes out identically whether speed, radius, and gravity are all expressed in metric (m/s, m, m/s²) or all in imperial (ft/s, ft, ft/s²) — no conversion between the two systems is actually needed for the angle itself, only for keeping each input's own displayed unit consistent with the others when the page's toggle is flipped.

FAQ

Do real roads actually use this exact angle?

Rarely at highway speeds — this is the angle that needs zero tire friction, but real pavement has plenty of friction available, so highway superelevation is deliberately built shallower than the full frictionless angle (commonly in the rough range of 4-8%, with the remaining cornering force supplied by friction) to avoid a slow-moving vehicle sliding down a steeply banked curve when stopped or crawling. Race tracks and velodromes, where speeds are much higher and more consistent, bank closer to or even beyond this ideal figure.

What happens if a vehicle takes the curve faster or slower than this speed?

At exactly the design speed, the bank angle alone balances the turn with no friction needed. Faster than that, the vehicle needs additional inward friction force to avoid sliding outward up the bank; slower than that, friction has to act the other way to keep it from sliding down the bank toward the curve's center — which is exactly why a steeply banked curve can feel like it's 'pulling' a slow-moving vehicle inward.

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