Bevel gear design guide: pitch cone angle, cone distance, face width and straight vs spiral bevel gears

What is a bevel gear, and how do you calculate the pitch cone angle?

Bevel gears have their teeth on a cone and transmit motion between intersecting shafts — most often at a shaft angle Σ of 90° — in hand drills, differentials, right-angle drives and robot joints. A pair behaves like two pitch cones rolling on each other with a common apex; the half-angle of each cone is the pitch cone angle δ, fixed by the tooth ratio.

i = z₂ / z₁　　for Σ = 90°: δ₁ = atan(z₁ / z₂), δ₂ = 90° − δ₁

For a 20:30 pair, δ₁ = atan(20/30) = 33.69° and δ₂ = 56.31°, with a ratio of 1.5. For shaft angles other than 90° use the general form:

tan δ₁ = sin Σ / (z₂/z₁ + cos Σ)　　δ₂ = Σ − δ₁

The same 20:30 pair gives δ₁ = 23.41° and δ₂ = 36.59° at Σ = 60°, and 40.89° / 79.11° at Σ = 120°. The bevel gear generator on this site is fixed at a 90° shaft angle; for other angles compute the cone angles with this formula and model the pair in CAD.

TIPParallel shafts call for spur or helical gears (the spur gear pair generator makes both at once); intersecting shafts for bevel gears; crossed, non-intersecting shafts for a worm drive or hypoid gears.

Key dimensions: cone distance R, face width b, outside diameter and virtual teeth

Module m and pitch diameter d of a bevel gear refer to the large (outer) end. Every tooth element converges to the apex, so teeth shrink towards it — which is why face width must be limited.

d = m · z　　R = m · √(z₁² + z₂²) / 2 = d₂ / (2 sin δ₂)　　b ≤ min(R/3, 10m)

Outside diameter da = m · (z + 2 cos δ)　　virtual teeth zv = z / cos δ　　small end / large end = (R − b) / R

The virtual tooth count zv is the tooth count of the equivalent spur gear on the developed back cone, used to judge undercut and tooth strength. At 20° pressure angle a gear undercuts when zv < 2/sin²α ≈ 17.1, so the minimum pinion teeth are about 17.1 · cos δ₁: 13 for 1:1, 16 for 1:2 and 17 for 1:3. Sizes for common ratios at m = 2:

At b = R/3 the small-end module is only two-thirds of the large-end module. For 20:30, m2, b = 10 mm: (36.06 − 10) / 36.06 ≈ 0.72, a small-end module of about 1.45. The axial height of the toothed part is b · cos δ — about 5.55 mm on the gear and 8.32 mm on the pinion.

TIPThe bevel gear generator warns when face width exceeds R/3 and refuses beyond R/2. Its spec cards list δ, R, da, zv and the recommended maximum face width, so you can check them against this table.

Straight, Zerol, spiral and hypoid bevel gears compared

Straight bevel gears have few teeth in mesh, so every tooth entering and leaving contact causes an impact — fine at low speed. Spiral bevel teeth are curved and always have more than one pair in contact, running much more smoothly, but the direction of their axial thrust depends on the spiral hand and the direction of rotation.

TIPWhen choosing spiral bevel gears, pick the hand so that under the main direction of rotation the pinion's thrust pushes it away from the apex, out of mesh. Thrust in the other direction pulls the gears tighter and can jam them. A mating pair always has opposite hands (left with right).

Miter gears, mounting and gear forces

Miter gears are a 1:1 bevel pair with δ₁ = δ₂ = 45°. Both gears are identical and turn the drive through 90° without changing speed. The "Miter m2 · 20:20" preset in the bevel gear generator is an example.

• The pitch cone apexes must coincide at the shaft intersection: the pinion's large-end pitch plane sits d₂ / 2 from the gear axis, and the gear's sits d₁ / 2 from the pinion axis — 30 mm and 20 mm for a 20:30, m2 pair.
• Mounting-distance errors cause tip interference (too close) or excessive backlash (too far); fine-tune the axial position of each gear with shims.
• Bevel gears always produce axial force, so both shafts need bearings, thrust washers or shoulders that can take thrust.
• Support the pinion on both sides (straddle mounting) where possible; if it must overhang, keep the bearings close to the gear and use a larger shaft to limit deflection.

Ft = 2T / dm　　dm = d − b · sin δ　　Fa = Ft · tan α · sin δ　　Fr = Ft · tan α · cos δ

Example: a 20:30, m2, b = 10 mm pinion driven at T = 1 N·m. With d = 40 mm and δ₁ = 33.69°, the mean pitch diameter is dm = 40 − 10 × sin 33.69° ≈ 34.45 mm, so Ft = 2 × 1000 / 34.45 ≈ 58.1 N; the pinion's axial force is Fa ≈ 11.7 N and its radial force Fr ≈ 17.6 N. At a 90° shaft angle the gear's axial force equals the pinion's radial force (17.6 N) and its radial force equals the pinion's axial force (11.7 N).

TIPFor multi-stage drives — a bevel stage followed by spur reduction, for instance — chain the stages in the gear ratio calculator to check the overall ratio, output torque and direction of rotation.

3D printing bevel gears

• Print large end (back face) down: the flanks lean in towards the apex, so like a pyramid each layer is smaller than the one below and no supports are needed.
• The larger the cone angle, the closer the flanks are to horizontal and the worse the layer stair-stepping. The gear of a 1:3 pair has δ₂ = 71.57°, so its flanks are only about 18° from the bed; use 0.1–0.12 mm layers or switch to SLA / MJF.
• Check the module at the small end: for FDM keep the small end ≥ m1 (SLA m0.5, SLS / MJF m0.8). At b = R/3 the small end is two-thirds of the large end, so FDM bevel gears want m1.5 or more at the large end.
• Use 0.1 mm backlash and 0.2 mm bore clearance on FDM; fix the gear to its shaft with a D-cut, or a set screw with a captive hex nut.
• Axial location matters more than with spur gears: use shoulders, shims or retaining rings so the apexes coincide and the gears cannot float axially.
• PETG or nylon with PTFE or silicone grease works well; for high loads or production runs, SLS / MJF nylon is the better choice.

TIPGenerate the gear first, then switch "Output part" to the pinion. Keep tooth counts, module and pressure angle identical for both so the cone angles match.