Banking Angle Calculator NZ

This calculator finds the ideal banking angle for a curved road or track, the angle at which a vehicle can round the bend at a given speed without relying on friction at all. When a vehicle corners, something must provide the inward centripetal force that bends its path. On a flat road that force comes entirely from friction between the tyres and the surface, which can run out in the wet or at speed. Banking the curve, tilting the road surface inward like a velodrome, a motor-racing track or a motorway on-ramp, lets a component of the vehicle's own weight provide that inward force instead. At the ideal banking angle for a particular speed and curve radius, the geometry is perfectly balanced: no friction is needed, the car simply follows the curve. This tool computes that angle. You enter the design speed and the radius of the curve, and the calculator returns the ideal banking angle, along with the angle in radians, the underlying ratio, and the maximum speed the banking alone supports. The results update as you type, so you can see how a tighter curve or a higher speed demands a steeper bank. Use it for physics problems, for understanding road and track design, or for motorsport and engineering interest. The formula is elegant: the ideal banking angle is the inverse tangent of the speed squared divided by the radius times gravity. A key insight is that the ideal angle is exact for one speed only; below it, a vehicle would tend to slide down the bank, and above it, outward, with friction making up the difference in practice. This is why banked curves are designed around a typical speed and still rely on some friction for the range of real traffic.

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32.5°
ideal banking angle
In radians0.567
v² / (r g)0.637
Design speed90 km/h

Ideal banking angle = inverse tangent of (speed² / (radius x gravity)). At this angle no friction is needed; it is exact for one speed only.

How it works

For a banked curve with no friction, the horizontal component of the normal force from the road provides the centripetal force, while the vertical component balances gravity. Dividing one by the other shows that the tangent of the banking angle equals the speed squared divided by the radius times gravity, so the angle is the inverse tangent of that ratio.

Worked example

For a design speed of 25 metres per second around a curve of radius 100 metres, the ratio is 25 squared divided by 100 times 9.81, which is 625 over 981, about 0.637. The ideal banking angle is the inverse tangent of 0.637, about 32.5 degrees. At that bank, a vehicle at 25 metres per second, around 90 kilometres per hour, needs no friction to corner.

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Frequently asked questions

What is a banking angle?

It is the angle a road or track is tilted inward on a curve. Banking lets part of a vehicle's weight provide the inward force needed to corner, reducing or removing the reliance on tyre friction. The ideal angle depends on the design speed and the radius of the curve.

How is the ideal banking angle calculated?

It is the inverse tangent of the speed squared divided by the radius times gravity. At this angle the horizontal part of the road's normal force exactly supplies the centripetal force, so a vehicle at the design speed needs no friction to follow the curve.

Does banking work at all speeds?

The ideal angle is exact for one speed only. Below it, a vehicle tends to slide down the bank; above it, outward. In practice friction makes up the difference across the range of real traffic speeds, which is why banked curves are designed around a typical speed.