Ackermann steering geometry is the foundational principle behind every car, kart, and truck that can corner without its front tyres scrubbing sideways. When a vehicle turns, the inner front wheel traces a smaller circle than the outer front wheel — so the two wheels must point at different angles to stay aligned with their respective circular paths. Get the angles wrong and at least one tyre is constantly fighting the direction of travel: this wastes energy, heats the rubber, and creates unpredictable handling.
This calculator applies the exact cotangent formula used by professional chassis engineers. Enter your wheelbase, track width, outer steer angle, and desired Ackermann percentage — and the tool instantly returns the ideal inner wheel angle, your actual inner angle, the scrub angle (how far off ideal you are), and all three turning radii, plus a live diagram of the geometry.
How it works
The geometry reduces to a clean trigonometric relationship. Place the vehicle on a flat plane. During a turn, all four wheels must rotate about a single instantaneous centre — a point that lies on a line extended through the rear axle. The two front wheels have different distances from that centre, so they need different steer angles.
The ideal Ackermann condition is expressed in cotangent form:
cot(delta_outer) − cot(delta_inner) = T / L
where T is the track width (tyre-centre to tyre-centre), L is the wheelbase (front axle to rear axle), delta_outer is the steer angle of the outer wheel, and delta_inner is the steer angle of the inner wheel. Rearranging:
delta_inner_ideal = arccot( cot(delta_outer) − T/L )
The turning radius to the midpoint of the rear axle is:
R = L / tan(delta_outer)
The Ackermann percentage linearly interpolates between 0% (parallel steer, both angles equal) and 100% (ideal inner angle). The actual inner angle at any given percentage is:
delta_inner_actual = delta_outer + (pct / 100) × (delta_inner_ideal − delta_outer)
The scrub angle is the absolute difference between the actual and ideal inner angles. A green result below 0.5° is excellent; amber between 0.5° and 2° is acceptable for road use; red above 2° indicates meaningful tyre scrub at that steer angle.
Worked example
A typical family hatchback: wheelbase 2,600 mm, track width 1,500 mm, outer steer angle 20°.
- cot(20°) = 1/tan(20°) ≈ 2.747
- T/L = 1500 / 2600 ≈ 0.577
- cot(delta_inner_ideal) = 2.747 − 0.577 = 2.170
- delta_inner_ideal = arccot(2.170) = atan(1/2.170) ≈ 24.74°
- Turning radius R = 2.6 / tan(20°) ≈ 7.14 m
- At 100% Ackermann: inner angle = 24.74°, scrub = 0.00°
- At 70% Ackermann: inner angle ≈ 22.32°, scrub ≈ 2.42°
| Wheelbase | Track | Outer angle | Ideal inner angle | Turning radius |
|---|---|---|---|---|
| 2,600 mm | 1,500 mm | 20° | 24.74° | 7.14 m |
| 2,600 mm | 1,500 mm | 10° | 11.73° | 14.76 m |
| 1,040 mm | 1,400 mm | 30° | 47.16° | 1.80 m |
| 3,200 mm | 1,800 mm | 15° | 18.31° | 11.95 m |
The third row is a typical 100 cc kart (very short wheelbase, wide track relative to length) — the huge 17° split between inner and outer angles is why karts need carefully shaped stub axles and tie-rod geometry.
Formula note
The cotangent form is preferred over the tangent approximation (tan delta_inner − tan delta_outer = T/L) because it is exact for all steer angles up to 90°. The tangent approximation is only accurate for small angles (below roughly 5°). At the 20° example above, the tangent approximation gives an inner angle of approximately 24.43°, an error of 0.31° — meaningful for precision chassis work.
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