Banked curves tilt the road so that part of the normal force pushes cars toward the centre of the turn. Racetracks, highway ramps and velodromes are banked; aircraft bank to turn for the same reason.
How to use the Angle of Banking Calculator
- What to calculate?
- Turn radius r
- Banking angle θ
- Speed v
- Friction
- Gravitational acceleration g
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose what to calculate. For speeds on a road, friction lets cars go faster than the ideal speed up to v_max (and prevents sliding down below v_min). For aircraft, there is no friction and the bank angle sets the turn radius.
Inputs explained
| Field | What to enter |
|---|---|
| What to calculate? | Choose one: Car: maximum and minimum speed on a banked curve, Ideal banking angle (no friction needed), Airplane: turn radius from bank angle |
| Turn radius r | Units: m, cm, mm, km, in, ft … |
| Banking angle θ | Units: °, rad, arcmin, arcsec, mrad |
| Speed v | Units: km/h, m/s, mph, knots, ft/s |
| Friction | Choose one: Tyre on dry asphalt, Tyre on wet asphalt, Tyre on snow, Tyre on ice … |
| Gravitational acceleration g | Units: m/s², ft/s², g, cm/s² |
What each result means
- Ideal speed (no friction) = sqrt([Turn radius r]×[Gravitational acceleration g]×[s]÷[c])
- Ideal banking angle = deg(atan([Speed v]^2÷([Turn radius r]×[Gravitational acceleration g])))
- Turn radius = [Speed v]^2÷([Gravitational acceleration g]×[s]÷[c])
- Load factor = 1÷[c]
- Time for a 360° turn = 2×pi×[Speed v]÷([Gravitational acceleration g]×[s]÷[c])
Example values
How the main result changes when Turn radius r changes and everything else stays at the example values above:
| Turn radius r (m) | Maximum speed (km/h) |
|---|---|
| 50 | 95.25 |
| 75 | 116.66 |
| 100 | 134.71 |
| 125 | 150.61 |
| 150 | 164.98 |
| 200 | 190.5 |
Formula
Ideal angle: tan θ = v²/(r·g)
v_max = √(r·g·(sin θ + μ cos θ)/(cos θ − μ sin θ)), v_min = √(r·g·(sin θ − μ cos θ)/(cos θ + μ sin θ))
Aircraft: r = v²/(g·tan θ), load factor n = 1/cos θ
Worked example
Inputs — What to calculate?: Car: maximum and minimum speed on a banked curve; Turn radius r: 100 m; Banking angle θ: 20 °; Friction: Tyre on dry asphalt; Gravitational acceleration g: 9.80665 m/s².
Results — Maximum speed = 134.71 km/h; Minimum speed = 0 km/h; Ideal speed (no friction) = 68.01 km/h.
Key terms
- Banking angle
- The tilt of a road or aircraft relative to horizontal.
Frequently asked questions
Why are curves banked?
So the normal force supplies part of the centripetal force, reducing reliance on friction.
What is the load factor in a 60° banked turn?
2 g – the occupants feel twice their weight.
Does a heavier car need a different angle?
No – mass cancels out.
What happens below the minimum speed?
On an icy steep bank, the car would slide down toward the inside.