Banked Curve Angle Calculator
A banked curve tilts the road or track so part of the surface's normal force points toward the centre of the turn, supplying the centripetal force. At one design speed the banking does all the work and no friction is needed. This calculator finds that ideal banking angle from the design speed and curve radius, using the inverse-tangent relation tan(theta) = v squared over (g times r). The gravitational acceleration is an editable input so you can use a precise local value. Mass does not appear: it cancels out.
Banked curve formula
tan(theta) = v^2 / (g * r)
theta = arctan( v^2 / (g * r) )
degrees = theta * 180 / pi
centripetal acceleration = v^2 / r
The relation comes from balancing the horizontal component of the normal force against the required centripetal force, with the vertical component balancing weight. Mass cancels, so the ideal angle depends only on speed, radius and gravity.
Reading the result
- Higher speed or tighter radius (smaller r) needs a steeper bank.
- At the design speed the ideal bank needs zero friction to hold the turn.
- Real roads add a friction allowance so a band of speeds is safe.
- Enter speed in metres per second; divide km/h by 3.6 or multiply mph by 0.44704 first.
Banked curve: frequently asked questions
What is the ideal banking angle of a curve?
The ideal (frictionless) banking angle is the angle at which a vehicle can round a curve at the design speed with no reliance on friction: the horizontal component of the normal force alone provides the centripetal force. It is given by the angle whose tangent equals v squared divided by (g times r).
What is the formula for a banked curve?
For the frictionless ideal: tan(theta) = v^2 / (g * r), where v is speed, g is gravitational acceleration (about 9.80665 m/s2), and r is the curve radius. The banking angle theta is the inverse tangent of that ratio. This calculator uses an editable g so you can use a precise local value.
Why bank a road or track?
Banking tilts the surface so part of the normal force points toward the centre of the turn, supplying centripetal force. At the design speed no friction is needed, which keeps the turn safe even in wet or icy conditions and reduces tyre wear. Real roads add friction margins for a range of speeds.
Does the vehicle's mass matter?
No. The ideal banking angle depends only on speed, radius and gravity, not mass: mass cancels out of the equation. A heavier vehicle needs more centripetal force, but it also presses harder on the surface, so the required angle is the same.
Official sources
- NASA Glenn Research Center: Speed, velocity and circular motion.
- U.S. National Institute of Standards and Technology: Standard acceleration of gravity.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.