Case 1
- Design speed
- 25m/s
- Curve radius
- 120m
27.97°
Open with these valuesBanked Curve Angle Calculator
27.97°
Result: 27.97 °Tilt a curve until the road's own normal force supplies the whole turning force and you need no grip at all: θ = arctan(v² ÷ (r × g)). A 120 m bend taken at 25 m/s wants 27.97°. Speed counts squared and the radius divides, so tight fast corners — velodromes — are banked steeply.
Worked examples
Case 2
- Design speed
- 10m/s
- Curve radius
- 100m
5.82°
Open with these valuesCase 3
- Design speed
- 40m/s
- Curve radius
- 300m
28.54°
Open with these valuesHow it's calculated
θ = arctan(v² ÷ (r × g))
- StepEnter the design speed in metres per second — divide km/h by 3.6.
- StepEnter the radius of the curve in metres.
- ResultRead the ideal banking angle in degrees above the horizontal.
Reference table
| Speed, radius | Where you see it | Banking angle (°) |
|---|---|---|
| 0, 120 | Standing still, flat road | 0.00 |
| 10, 100 | Slow city bend | 5.82 |
| 30, 200 | Motorway ramp | 24.65 |
| 15, 50 | Same v²/r, half the radius | 24.65 |
| 25, 120 | Fast sweeping bend | 27.97 |
| 40, 300 | High-speed rail curve | 28.54 |
Questions
How do I calculate the ideal banking angle?
Take the inverse tangent of the speed squared divided by the radius times gravity: θ = arctan(v² / (r × g)), with g = 9.80665 m/s². Use metres per second for the speed and metres for the radius to get the angle in degrees. A 120 m curve at 25 m/s gives arctan(625 / 1176.8) = 27.97°.
What is a banked curve?
A banked curve is a road, track, or railway curve whose surface is tilted inward, so part of the road's normal force points toward the centre of the curve. That inward component supplies centripetal force, helping a vehicle turn without relying on tyre friction. You see banking on motorway ramps, velodrome tracks, and high-speed rail curves.
Why does the ideal angle need no friction?
At the ideal angle the horizontal component of the road's normal force exactly equals the centripetal force a vehicle needs at the design speed. Because that geometry alone supplies the turning force, no friction is required — friction only becomes necessary above or below the design speed.
How do speed and radius change the angle?
Speed is squared in the formula and the radius sits in the denominator, so both a faster design speed and a tighter curve raise the required angle. Double the speed and the tangent quadruples; halve the radius and it doubles. That is why sharp, high-speed bends like velodrome corners are banked so steeply.
What units should I use?
Use SI units: metres per second for the speed and metres for the radius, which gives the banking angle in degrees. If your speed is in km/h, divide by 3.6 first. The calculator uses standard gravity, g = 9.80665 m/s².
Sources and last check
Information, not professional advice.