Straight bevel gears transmit rotation between intersecting shafts. Their teeth sit on pitch cones, so the geometry depends on the tooth counts, shaft angle, module convention and face width. The equations here cover a basic external pair with axes at 90 degrees and module measured at the large end. They help check preliminary dimensions; they are not a tooth-strength rating or a manufacturing drawing.

Bevel gear dimension formulas

Inputs and units

Use z₁ for the pinion tooth count and z₂ for the gear tooth count. The ratio i = z₂/z₁ is dimensionless. For metric work, module m is in millimetres, and a pitch diameter calculated as m × z is in millimetres. The pitch-cone angles δ₁ and δ₂ describe each member’s cone relative to its axis. On a 90-degree external pair, those angles add to 90 degrees. If the shaft angle is not 90 degrees, do not use the simplified angle relationship in this table.

The module convention matters. Autodesk’s bevel-gear geometry documentation distinguishes the module at the outer cone from dimensions at the mean cone. If a drawing or gear catalog gives a module at another reference section, convert it using the applicable gear geometry method before combining it with these equations. Mixing outside and mean-cone dimensions creates inconsistent tooth sizes and center geometry.

Worked geometry example

Consider a 20-tooth pinion driving a 40-tooth gear on intersecting 90-degree shafts. Let the outside module be 2 mm. The ratio is 40/20 = 2:1. The outside pitch diameters are 2×20 = 40 mm and 2×40 = 80 mm. The pinion pitch-cone angle is arctan(20/40), about 26.565 degrees; the gear angle is about 63.435 degrees. Their sum is 90 degrees. The outer cone distance is one-half of the square root of 40²+80², or about 44.72 mm.

If the face width is 10 mm, the mean cone distance is 44.72−5 = 39.72 mm. The mean-cone module is 2×39.72/44.72, about 1.776 mm. These are geometric checks. They do not determine backlash, root clearance, tooth contact, bearing reactions or allowable torque.

Choose the correct geometry

Confirm the tooth counts, shaft intersection angle, outside module or diametral-pitch convention, pressure angle and whether the gear is straight, spiral or hypoid. Bevel gear software may also require tooth profile, face width, addendum, dedendum, profile shift and tooth-thickness correction. Match the same reference cone and unit system across both members. A pair with the right ratio can still have incompatible tooth forms.

Common mistakes

First, do not use a spur-gear pitch diameter at an unspecified cone section as if it were the bevel gear’s outside diameter. Second, do not assume equal 45-degree pitch angles unless the tooth counts are equal. Third, do not treat a geometric result as a load-capacity calculation. For a production design, follow the selected gear standard or manufacturer method and verify strength, contact pattern, lubrication, alignment and tolerances. The equations are based on the basic geometry inputs and parameter definitions described in Autodesk’s bevel-gear geometry reference; consult the applicable current standard for detailed design.

Before turning the example into a drawing, define which member is the pinion and which is the gear, then keep those labels consistent in the tooth counts and angles. Confirm that both gears use the same module at the same reference cone and that the shaft axes intersect at the assumed angle. If the gear pair uses a nonstandard shaft angle or offset, the right-triangle construction and complementary-angle result no longer apply. A CAD model can help visualize the pitch cones, but a plausible-looking model does not verify tooth contact or strength.

For a released design, record the tooth system, pressure angle, material, heat treatment, face width, backlash target, load spectrum and lubrication. Use a gear rating method suitable for the geometry and duty cycle, and check the manufacturer’s tolerances for the actual cutting process. This compact formula list is for preliminary geometry and unit checks only.