Bank Angle Calculator

Find the bank angle needed for a level, coordinated turn at a given speed and turn radius, plus the resulting load factor and turn rate.

Results

Calculated
Bank angle
—
θ = arctan(v² ÷ (g·r))
Load factor
—
n = 1 ÷ cos(θ)
Turn rate
—
Angular speed of the turn
Time for full 360° turn
—
T = 2πr ÷ v

How the bank angle calculation works

When an aircraft or vehicle turns at constant speed and constant altitude without skidding, the horizontal (centripetal) force that bends its path has to come from somewhere. Banking tilts the lift force (or, for a vehicle, the normal force from the road) so that part of it points sideways, toward the center of the turn, instead of purely upward. The steeper the bank, the larger that sideways component becomes.

The formula

For a level, coordinated turn at speed v and turn radius r, under gravitational acceleration g, balancing the vertical and horizontal force components gives the standard bank angle equation:

tan(θ) = v² ÷ (g × r), so θ = arctan(v² ÷ (g × r))

This calculator converts the entered speed from km/h to m/s, applies the formula to get the bank angle in degrees, then derives two related quantities. The load factor, n = 1 ÷ cos(θ), is how many times the object's normal weight the supporting structure must bear during the turn — it is what pilots and drivers feel as extra "g-force" in a turn. The turn rate is the angular speed of the turn itself, v ÷ r in radians per second, and the time for a full 360° turn is the circumference of the turn circle divided by speed, 2πr ÷ v.

Interpreting the results

A shallow bank (under about 20°) keeps the load factor close to 1 and feels gentle. As the bank angle climbs toward 60°, the load factor doubles to about 2g, and it keeps rising sharply after that — a bank angle at or beyond 90° would require infinite lift and is not achievable in level flight, which is why very high speeds combined with tight radii are flagged as invalid. A larger radius or a lower speed always reduces the required bank angle for the same turn.

Frequently Asked Questions

What is the bank angle formula?
For a level, coordinated turn that relies purely on banking rather than friction, the required bank angle satisfies tan(θ) = v² ÷ (g × r), where v is speed, r is the turn radius, and g is gravitational acceleration. Solving for the angle gives θ = arctan(v² ÷ (g × r)).
Why does a turn need to be banked at all?
In a banked turn, part of the force that normally supports weight is tilted sideways, and that sideways component supplies the centripetal force needed to curve the path. On a flat, unbanked turn taken at speed, only friction can supply that sideways force, so banking lets a tighter or faster turn be flown or driven with far less reliance on friction.
What is load factor in a banked turn?
Load factor is the ratio of the supporting force during the turn to the object's weight, given by n = 1 ÷ cos(θ). At a 60° bank the load factor is 2, meaning the structure must support twice the normal weight. Load factor rises sharply as the bank angle approaches 90°.
How do speed and radius affect the required bank angle?
Bank angle increases with speed and decreases with a larger turn radius, since tan(θ) is proportional to v squared and inversely proportional to r. Doubling the speed roughly quadruples v squared, so a much steeper bank angle (or a much larger radius) is needed to hold the same turn.