Calculate ideal banking angle, velocity, radius, or gravity for a banked curve, plus a theoretical no-slip speed range using an entered friction coefficient.

Steady circular motion on a banked surface. Friction results are theoretical limits, not safe driving speeds; rollover, tire variation and road conditions are omitted.

9.81 m/sยฒ is an Earth-surface approximation.

Ideal Banking Angle Formula

The following formula is used to calculate the ideal banking angle for a given velocity, radius, and gravitational acceleration.

ฮธ = atan((vยฒ) / (r ร— g))

Variables:

  • ฮธ is the inward banking angle from the horizontal; convert an arctangent in radians to degrees if needed
  • v is the velocity in meters per second (m/s)
  • r is the radius of the curve in meters (m)
  • g is the gravitational acceleration in meters per second squared (m/sยฒ)

To calculate the ideal banking angle, divide the square of the velocity by the product of the radius and gravitational acceleration, then take the arctangent of the result.

What is an Ideal Banking Angle?

The ideal banking angle is the angle at which a road or track should be banked (tilted) to allow vehicles to travel around a curve at a specific speed without relying on friction. In this steady circular-motion model, the normal force supplies the required centripetal force without lateral tire friction. The calculation does not establish a safe driving speed and omits rollover, tire variation and road conditions. The ideal banking angle depends on the speed of the vehicle, the radius of the curve, and the gravitational acceleration.

How to Calculate Ideal Banking Angle?

The following steps outline how to calculate the Ideal Banking Angle.

  1. First, determine the velocity of the vehicle (v).
  2. Next, determine the radius of the curve (r).
  3. Next, determine the gravitational acceleration (g), typically 9.81 m/sยฒ.
  4. Finally, calculate the ideal banking angle using the formula ฮธ = atan((vยฒ) / (r * g)).
  5. After inserting the values and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Velocity (v) = 20 m/s

Radius (r) = 50 m

Gravitational Acceleration (g) = 9.81 m/sยฒ