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Banked Curve Speed Calculator

Enter the curve radius and bank angle to get the ideal speed.

How it is calculated

The tool uses v = √(r × g × tan θ), where r is the curve radius in metres, θ is the bank angle in degrees (converted to radians for the tangent) and g = 9.80665 m/s². The result v is the speed at which the road's normal force alone supplies the centripetal force, with no tire friction needed. It is shown in m/s, then converted with ×3.6 to km/h and ×2.23694 to mph. The vehicle mass cancels out of the equation, so it is not asked for. The radius must be above zero and the angle strictly between 0 and 90 degrees; an angle of 0 (flat road) or 90 gives no result. Real roads rely on friction too, so this is the design speed, not a safe maximum.

Worked example

A curve with a radius of 100 m banked at 10°. tan 10° = 0.1763. Then r × g × tan θ = 100 × 9.80665 × 0.1763 = 172.9, and the square root is 13.15 m/s. Converting: 13.15 × 3.6 = 47.34 km/h, or 29.42 mph. A car at this speed needs no sideways friction to follow the curve; faster, and it relies on grip to avoid sliding up the bank.

Try this example →

Common mistakes

  • Entering the bank angle as a percentage grade (a 10% slope is about 5.7°, not 10°) or in radians. The field is degrees.
  • Using the diameter of the curve instead of its radius. Doubling r raises the speed by a factor of 1.41, not 2.
  • Expecting a heavier vehicle to have a different ideal speed. Mass cancels out; only radius and angle matter.
  • Treating the result as the legal or safe maximum. It is the speed at which friction is not needed; with tire grip, vehicles can safely go faster, and on ice or wet roads less friction is available.

About this calculator

A banked curve tilts the road surface inward so that part of the normal force from the road points toward the center of the turn, helping supply the centripetal force needed to keep a vehicle moving in a circular path. At the ideal speed for a given bank angle, the horizontal component of the normal force alone provides exactly enough centripetal force — no friction from the tires is needed at all, which is why this speed is also called the "no-friction" or design speed of the curve.

This calculator finds that ideal speed from the curve's radius and bank angle using v = √(r·g·tanθ), derived by setting gravity, the normal force, and the required centripetal force in balance. It's the formula behind how civil and highway engineers set banking angles on highway ramps and racetracks, and it's a standard example in introductory physics courses covering circular motion and Newton's laws.

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