Calculate the ideal outer wheel angle, turning radius, and Ackermann percentage from your wheelbase, track width, and inner steering angle.
Ackermann Steering Angle Formula
The Ackermann condition sets the outer wheel angle that lets both front tires roll around one shared turn center. It compares the cotangents of the inner and outer steer angles to the ratio of track width and wheelbase.
cot(δo) = cot(δi) + T / L
Solved for the outer wheel angle, that becomes:
δo = atan( 1 / ( cot(δi) + T / L ) )
The turning radius to the center of the rear axle follows from the inner wheel angle, the wheelbase, and the track width:
R = T / 2 + L / tan(δi)
When you already know the outer angle a linkage produces, the Ackermann percentage measures how close that design sits to the ideal:
Ackermann % = 100 x (δi - δo) / (δi - δo,ideal)
- δi = inner (near-side) front wheel steering angle, in degrees
- δo = outer (far-side) front wheel steering angle, in degrees
- δo,ideal = outer angle from perfect Ackermann geometry
- T = front track width, the distance between the two steering axes (kingpins)
- L = wheelbase, the distance from the front axle to the rear axle
- R = turning radius measured to the center of the rear axle
The calculator runs three functions from these relationships. The ideal outer angle mode takes your wheelbase, track, and inner wheel angle and returns the outer angle for scrub-free geometry, along with the toe-out split, the Ackermann ratio, and the turning radius and turning circle. The Ackermann percentage mode adds the outer angle your linkage actually produces and reports what fraction of true Ackermann it reaches, flagging parallel, partial, near-true, or anti-Ackermann setups. The target radius mode reverses the math: you give it the turning radius you need and it returns the inner and outer lock angles required to reach it.
Typical Ackermann Percentage by Vehicle Type
Real linkages rarely run 100 percent Ackermann across the whole steering range, because the trapezoidal arm layout only matches the ideal curve at one steer angle. The target percentage depends on how fast the vehicle corners and how much tire scrub costs it. Use the design ranges below to judge your own result.
| Vehicle type | Typical Ackermann % | Why |
|---|---|---|
| Passenger car | 50 to 80% | Balances parking-lot scrub against high-speed stability |
| Forklift or warehouse truck | 90 to 100% | All maneuvering is low speed, so scrub is the main tire cost |
| Airport tug or ground tractor | 80 to 100% | Tight 90 degree turns around aircraft stands |
| Row-crop tractor | 85 to 100% | Sharp headland turns without tearing the soil |
| Go-kart or simple trailer | Near 0% (parallel) | Short wheelbase and low cost favor a plain layout |
| Formula SAE or race car | Negative (anti) | The loaded outer tire needs more slip angle than the inner |
The next table shows how the ideal outer angle and toe-out grow with steering input for a mid-size car with a 2.7 m wheelbase and 1.5 m front track. At small angles the two wheels turn almost together, which is why Ackermann barely matters for gentle highway curves and starts to pay off only near full lock.
| Inner wheel angle | Ideal outer angle | Toe-out (inner minus outer) |
|---|---|---|
| 10 degrees | 9.1 degrees | 0.9 degrees |
| 20 degrees | 16.9 degrees | 3.1 degrees |
| 30 degrees | 23.6 degrees | 6.4 degrees |
| 35 degrees | 26.8 degrees | 8.2 degrees |
| 40 degrees | 29.8 degrees | 10.2 degrees |
Example Problems
Example 1: find the ideal outer angle. A car has a wheelbase of 2.5 m and a front track of 1.4 m. At full lock the inner wheel turns 35 degrees. Take the cotangent of the inner angle, cot(35) = 1.428. Add the track-to-wheelbase ratio, 1.4 / 2.5 = 0.56, to get 1.988. The outer angle is atan(1 / 1.988) = 26.7 degrees. The two wheels differ by 8.3 degrees, and the turning radius to the rear axle center is 0.7 + 2.5 / tan(35) = 4.27 m.
Example 2: work back from a target radius. You need a delivery vehicle with a 2.6 m wheelbase and 1.5 m track to turn inside a 6 m radius measured to the rear axle center. The inner arm length is 6 minus 0.75 = 5.25 m, so the inner wheel must reach atan(2.6 / 5.25) = 26.35 degrees. The outer wheel needs atan(2.6 / 6.75) = 21.07 degrees, a split of 5.3 degrees. Those are the full-lock angles your steering stops and linkage have to allow.
Frequently Asked Questions
What is a good Ackermann percentage? It depends on speed. Road cars usually run 50 to 80 percent because they trade a little low-speed scrub for stability at speed. Low-speed machines like forklifts and airport tugs run close to 100 percent, where clean rolling saves the tires. Race cars often run anti-Ackermann, a negative percentage, because the heavily loaded outer tire generates peak grip at a larger slip angle than the lightly loaded inner tire.
Why does the inner wheel turn more than the outer wheel? In a turn the inner wheel follows a tighter circle than the outer wheel, and a tighter circle needs a sharper steer angle. If both wheels turned the same amount, one of them would have to slide sideways to follow its path, and that sliding is the scrub you hear as a squeal at walking pace.
Does Ackermann geometry change the turning radius? Not on its own. The turning radius is set by the inner wheel angle, the wheelbase, and the track width, which is why the target radius mode needs only those inputs. Ackermann geometry decides the outer wheel angle, which controls whether that wheel rolls cleanly or scrubs, not how tight the circle is.
