# Angle of Banking Calculator

URL: https://speedycalculator.com/calculators/physics/bank-angle/
Last updated: 2026-10-30

> The angle of banking calculator finds the ideal banking angle of a curve, the maximum and minimum safe speeds on a banked road with friction, or an aircraft's turn radius and load factor from its bank angle.

## Inputs

- What to calculate?
- Turn radius r (m, cm, mm, km, in, ft, yd, mi)
- Banking angle θ (°, rad, arcmin, arcsec, mrad)
- Speed v (km/h, m/s, mph, knots, ft/s)
- Friction
- Gravitational acceleration g (m/s², ft/s², g, cm/s²)

## Results

- Maximum speed
- Minimum speed
- Ideal speed (no friction)
- Ideal banking angle
- Turn radius
- Load factor
- Time for a 360° turn

## 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
```

## Explanation

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

FieldWhat to enterWhat to calculate?Choose one: Car: maximum and minimum speed on a banked curve, Ideal banking angle (no friction needed), Airplane: turn radius from bank angleTurn radius rUnits: m, cm, mm, km, in, ft …Banking angle θUnits: °, rad, arcmin, arcsec, mradSpeed vUnits: km/h, m/s, mph, knots, ft/sFrictionChoose one: Tyre on dry asphalt, Tyre on wet asphalt, Tyre on snow, Tyre on ice …Gravitational acceleration gUnits: 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)5095.2575116.66100134.71125150.61150164.98200190.5

## 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.

## FAQ

**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.


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)