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Banked Curve Calculator

Find the banking angle a curve needs at a given speed.

How it is calculated

The tool gives the ideal banking angle with no friction: θ = arctan(v² / (r × g)). The radius r is in metres and the speed is entered in km/h, which the tool converts to m/s by dividing by 3.6. g is fixed at 9.81 m/s². It also shows the centripetal acceleration the vehicle needs, a = v² / r, in m/s², and the angle in degrees.

At this angle the horizontal part of the normal force from the road supplies the whole turning force, so the vehicle needs no sideways friction. The radius must be greater than zero and the speed cannot be negative. The mass of the vehicle does not appear, because it cancels out. The result is for the design speed only: at other speeds friction or a different banking is needed, and real roads add friction, so it is not a full road-design check.

Worked example

A curve of radius 100 m designed for 80 km/h. Convert the speed: 80 / 3.6 = 22.22 m/s. Then v² = 493.8 and v² / (r × g) = 493.8 / (100 × 9.81) = 0.503. The angle is arctan(0.503) = 26.72°. The centripetal acceleration is 493.8 / 100 = 4.94 m/s², about half of g. A wider 250 m curve at 90 km/h needs only 14.30°.

Try this example →

Common mistakes

  • Entering speed in m/s or mph. The box is km/h, so 22 m/s typed as 22 would be treated as a slow 6.1 m/s and give a much smaller angle.
  • Entering the curve's diameter or its length instead of the radius. The radius is measured from the centre of the turn.
  • Treating the angle as a safe limit for any speed. It is exact for one speed only; faster needs friction, slower may have the vehicle slide down the bank.
  • Confusing the centripetal acceleration with a number in g. It is shown in m/s², so divide by 9.81 to compare it with g.
  • Expecting the vehicle's weight to matter. Mass cancels in the equation, so a car and a truck have the same ideal angle.

About this calculator

This calculator finds the ideal (frictionless) banking angle for a curve at a given radius and design speed, using θ = arctan(v²/rg), where v is speed, r is the curve's radius, and g is gravitational acceleration. At this angle, the horizontal component of the normal force alone supplies the centripetal force needed to turn, with no reliance on friction — meaning a vehicle could take the curve at that exact speed even on ice.

Civil and highway engineers use this formula when designing banked roads, racetrack turns, and railway curves to set a safe superelevation for the intended speed, since taking a curve too fast for its banking pushes a vehicle outward (relying on friction to compensate) while taking it too slow can pull it toward the inside edge.

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