Banked Curve Calculator
Enter a speed and a curve radius to get the ideal banking angle in degrees — the tilt that lets a vehicle round the bend with no friction at all.
The friction-free angle
A banked curve calculator returns the angle θ = arctan(v²/rg) at which the road can be tilted so a car holds the curve without relying on tyre grip.
Use SI units
Speed in metres per second and radius in metres give the angle in degrees — divide km/h by 3.6 to get m/s before you start.
What is the ideal banking angle?
The tilt that needs no friction
The ideal banking angle is the angle at which you would tilt a curved road so that a vehicle travelling at one chosen design speed can round the bend without any friction at all — gravity and the normal force from the road do all the work of turning the car. A banked curve calculator turns two measurements, the design speed in metres per second and the radius of the curve in metres, into that angle in degrees. It is the geometry behind highway interchange ramps, velodrome tracks, and railway curves, where engineers tilt the surface inward so that traffic stays on the curve smoothly even when the road is wet or icy and grip is unreliable.
Enter a speed in metres per second and a curve radius in metres to get the ideal banking angle in degrees instantly.
The ideal banking angle is the inverse tangent of the speed squared divided by the radius times the gravitational acceleration.
θ = arctan(v² / (r × g))The speed is squared and the radius sits in the denominator, so faster speeds and tighter curves both call for a steeper bank. The gravitational acceleration g is the standard value 9.80665 m/s². Use metres per second for the speed and metres for the radius and the angle comes back in degrees.
Suppose a highway ramp has a radius of 120 m and is designed for a speed of 25 m/s (about 90 km/h).
Square the speed
25² = 625 — the squared speed that drives the angle.
Multiply the radius by gravity
120 × 9.80665 = 1176.798 — the radius times g in the denominator.
Take the inverse tangent
arctan(625 / 1176.798) = arctan(0.5312) = 27.97° — the ideal banking angle.
The angle (27.97° for the ramp above) is the tilt at which a vehicle travelling at exactly the design speed rounds the curve needing no friction at all — the horizontal component of the road's normal force supplies precisely the centripetal force required, so the car neither slides up nor down the bank. Faster speeds or tighter curves push the angle up: double the speed and the required tangent quadruples, while halving the radius doubles it, so a sharp, high-speed bend can demand a dramatic bank. Below the design speed a real road still works because friction makes up the small shortfall, and above it friction supplies the extra grip — the ideal angle simply marks the single speed at which neither is needed. This is why a velodrome's steepest banking sits where riders move fastest and why motorway slip roads tilt more sharply the tighter they curve.
The formula is exact for the idealised case, but a couple of practical points are worth keeping in mind.
Frictionless ideal at one design speed
This calculator gives the frictionless ideal banking angle: it assumes no tyre friction and a single design speed, so a real road designed for a range of speeds is banked using a friction allowance rather than this bare angle. It models a level, uniformly banked curve and ignores camber transitions, road crown, and vehicle dynamics. Keep your units consistent — metres per second for speed and metres for the radius — or the angle will be wrong: convert km/h to m/s by dividing by 3.6 before you enter the speed.