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Banking Angle Calculator

Find the ideal banking angle of a curve.

How it is calculated

The tool gives the angle at which a banked curve needs no friction at all: θ = arctan(v² ÷ (r × g)), with speed v in metres per second, curve radius r in metres and g = 9.81 m/s². The angle is shown in degrees with two decimals.

It is the ideal angle for one specific speed. Below that speed a vehicle tends to slide down the slope and needs friction to stay up; above it the vehicle tends to slide outward. Radius must be greater than zero and speed cannot be negative. Speed 0 gives 0°. The tool treats the vehicle as a point on a frictionless surface, so it does not include tyre grip, road crossfall limits or how steep a road may legally be built. To convert km/h to m/s, divide by 3.6.

Worked example

A curve of radius 200 m taken at 90 km/h. Convert the speed: 90 ÷ 3.6 = 25 m/s.

v² = 25² = 625. r × g = 200 × 9.81 = 1962. Ratio: 625 ÷ 1962 = 0.3186. Angle: arctan(0.3186) = 17.67°. A frictionless vehicle would hold the curve at 90 km/h on a bank this steep; real roads are banked far less and rely on tyre grip for the rest.

Try this example →

Common mistakes

  • Typing the speed in km/h or mph. The formula needs m/s, and a raw 90 would give about 76.4° instead of 17.67°.
  • Entering the diameter instead of the radius. A doubled r roughly halves the tangent of the angle.
  • Entering the radius in feet or kilometres while the speed is in m/s. Both must use metres and seconds.
  • Thinking a higher speed needs the same bank. The ideal angle grows with the square of the speed, so doubling it roughly quadruples the tangent.
  • Reading the result as a safe design angle. It is the friction-free ideal for one speed only, not a road-design limit.

About this calculator

This calculator finds the ideal banking angle of a curved road or racetrack — the angle at which a vehicle can take the turn at a given speed with zero reliance on friction — using θ = atan(v²/(r·g)), derived by setting the horizontal component of the normal force equal to the centripetal force required for circular motion (v²/r) while gravity g balances the vertical component.

Real roads are banked (superelevated) close to this ideal angle for their design speed so that tires experience less side-load, reducing wear and the risk of skidding, especially in rain or ice; racetracks like NASCAR ovals use much steeper banking to let cars corner at far higher speeds than a flat track would allow.

Civil and highway engineers designing curve superelevation, motorsport engineers analyzing track design, and physics students studying circular motion and centripetal force all use this formula to relate speed, radius, and banking angle.

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