Generated on: June 17, 2026 at 05:24 PM

Banking Angle Calculator

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Design speed90 km/h
Turn radius100 m
Gravity9.80665 m/s²

Generated with OneCalc · https://onecalc.app/en/physics/banking-angle

Physics

Banking Angle Calculator

Find the ideal banking angle for a curve with tan θ = v²/(rg). Enter the speed and turn radius to get the angle a road or track should be tilted so a vehicle can round it without relying on friction.

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Banking Angle

Watch a velodrome, a motorway interchange or a railway curve and you will see the surface tilted inward. That tilt is the banking angle, and it is chosen so a vehicle at the design speed can hold the curve without depending on friction at all. This calculator finds the ideal angle from the speed and the radius of the turn.

Where the formula comes from

To follow a curve, a vehicle needs a net inward (centripetal) force of mv2/rmv^2/r. On a flat road only sideways friction can supply it. On a banked surface, the road pushes perpendicular to itself, and tilting it means part of that normal force now points inward. Balancing the inward component of the normal force against the centripetal requirement, and the vertical component against gravity, the mass cancels and leaves tan⁡θ=v2/(rg)\tan\theta = v^2/(rg).

Formula

QuantitySymbolMeaning
Banking angleθ\thetaθ=arctan⁡ ⁣(v2rg)\theta = \arctan\!\left(\dfrac{v^2}{rg}\right)
Centripetal accelerationaaa=v2ra = \dfrac{v^2}{r}
Design speedvvSpeed the bank is built for
RadiusrrRadius of the curve
Gravitygg9.80665 m/s29.80665\ \text{m/s}^2

Worked example

A curve of radius r=100 mr = 100\ \text{m} designed for v=25 m/sv = 25\ \text{m/s} (90 km/h):

θ=arctan⁡ ⁣(v2rg)=arctan⁡ ⁣(252100×9.80665)=arctan⁡(0.637)≈32.5∘\begin{aligned} \theta &= \arctan\!\left(\frac{v^2}{rg}\right) \\ &= \arctan\!\left(\frac{25^2}{100 \times 9.80665}\right) \\ &= \arctan(0.637) \approx 32.5^\circ \end{aligned}

The centripetal acceleration is a=v2/r=625/100=6.25 m/s2a = v^2/r = 625/100 = 6.25\ \text{m/s}^2, about 0.64 g.

Why mass drops out

A heavier vehicle needs a larger inward force, but it also presses harder on the road and so receives a proportionally larger normal force. The two effects cancel, which is why the same banked curve works for a motorcycle and a loaded truck alike — the ideal angle depends only on speed, radius and gravity.

Limitations

The result is the ideal angle for one design speed, where friction is not needed at all. Drive slower and the vehicle tends to slip down the bank; drive faster and it tends to slide up and out — in both cases friction makes up the difference, which is why real curves work over a range of speeds. The formula assumes a point vehicle on a uniformly banked, rigid surface and ignores aerodynamic forces, camber changes and the height of the centre of mass (which matters for rollover but not for the ideal angle).

Frequently Asked Questions (FAQ)

What is the banking angle formula?

The ideal banking angle satisfies tan θ = v²/(rg), so θ = atan(v²/rg), where v is the design speed, r the radius of the curve and g gravity. At this angle the horizontal component of the normal force supplies exactly the centripetal force needed, so the vehicle holds the curve even on a frictionless surface.

Why are roads and tracks banked?

On a flat curve, only sideways friction keeps a vehicle from sliding outward. Tilting the surface lets a component of the normal force point inward instead, doing part or all of that job. This is why highway interchanges, velodromes and railway curves are banked — it lets vehicles corner faster and more safely, and reduces wear on tyres and rails.

Does the vehicle's mass change the angle?

No. Mass cancels out of the equation, so the ideal banking angle depends only on the speed, the radius and gravity — not on whether a motorcycle or a loaded truck is taking the curve. A heavier vehicle needs a larger force, but it also has proportionally more weight to provide it.

What happens at other speeds?

The banking angle is ideal for one design speed. Below it, a car tends to slide down the bank and friction must hold it up; above it, the car tends to slide up and outward and friction must hold it in. Real banked curves are set for a typical speed, and friction covers the range of speeds around it.

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