Bevel Gear Pitch Angle⚠ unverified
Mechanical / Gears · Compute the pitch cone angle of a bevel gear
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| z1 | z1 | — | 1.0 | Number of teeth on the gear whose pitch angle is computed |
| z2 | z2 | — | 1.0 | Number of teeth on the mating gear |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | γ | — | Pitch angle, in degrees |
The science & history
Understanding the Parameters
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Tooth counts $z_1$, $z_2$ — their ratio sets the pitch angle because the gear ratio equals the tooth-count ratio and also the ratio of the pitch-cone radii. A gear with more teeth than its mate ($z_1 > z_2$) gets the larger cone angle ($\gamma > 45^\circ$); an equal pair each get 45°.
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The complementary pair — for a 90° pair, the two pitch angles satisfy $\gamma_1 + \gamma_2 = 90^\circ$, so computing one gives the other by subtraction. The pinion (fewer teeth) is the steep, pointed cone; the gear (more teeth) is the shallow, wide cone.
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Why it matters — the pitch angle sets the entire cone geometry: the back-cone used to define tooth proportions, the mounting angles, and the direction of the separating and thrust forces that the bearings must react (bevel gears, like helical, produce axial thrust).
Derivation (Approaching a Proof)
Two bevel gears mesh along a common line where their pitch cones touch, both cones sharing the apex at the shaft intersection. At any radius, the two gears roll without slipping, so their pitch-cone radii are in the same ratio as their tooth counts and their speeds:
$$\frac{r_1}{r_2} = \frac{z_1}{z_2}.$$
For a 90° shaft angle, the two pitch-cone axes are perpendicular, and the shared pitch line makes angle $\gamma_1$ with gear 1's axis and $\gamma_2 = 90^\circ - \gamma_1$ with gear 2's axis. The pitch radii project onto the shared line such that
$$\tan\gamma_1 = \frac{r_1}{r_2} = \frac{z_1}{z_2} \;\Longrightarrow\; \gamma_1 = \arctan\frac{z_1}{z_2}.$$
Geometrically, the tangent of the cone half-angle is the "opposite over adjacent" of the two perpendicular pitch radii. The complementary gear angle follows from $\gamma_1 + \gamma_2 = 90^\circ$. For a general shaft angle the perpendicular projection is replaced by the law-of-sines form in the registry note.
Dimensional check. $z_1/z_2$ is a dimensionless ratio; $\arctan$ of it is an angle. ✓
History and Development
Bevel gears are ancient in concept (right-angle drives) and were put on a rigorous involute/octoid tooth basis in the 19th–20th centuries; spiral bevel and hypoid gears (Gleason, early 1900s) added smoother, stronger curved teeth used in every car's differential. The pitch-cone geometry, with $\gamma = \arctan(z_1/z_2)$ for right-angle pairs, is standard in Shigley and AGMA bevel-gear practice.
Related Concepts: Gear Bending Stress Lewis, Gear Contact Stress, Helical Gear Thrust, Gear Power Capacity, Planetary Ratio
Notes: Pitch (cone) half-angle; output is an angle (deg) despite the registry label. Assumes 90° shaft angle ($\gamma_1 + \gamma_2 = 90^\circ$); use the law-of-sines form for other shaft angles.