Helical gears can be described in a normal plane, measured perpendicular to the teeth, or a transverse plane, perpendicular to the gear axis. Mixing those two systems is a common source of incorrect pitch diameter and pressure-angle calculations. This compact table collects basic relationships for standard, unshifted helical gears.

Keep the normal and transverse systems distinct

The normal module mₙ and normal pressure angle αₙ describe geometry in the plane perpendicular to the tooth direction. The transverse module mₜ and transverse pressure angle αₜ describe geometry in the cross-section perpendicular to the shaft axis. The reference helix angle β links the two. When β is zero, the geometry reduces to the corresponding spur-gear relationship; as β increases, the transverse module is larger than the normal module.

For a standard helical gear, convert module using mₜ = mₙ / cos β. The transverse pressure angle follows tan αₜ = tan αₙ / cos β. Use the same angle convention and units throughout the calculation. Trigonometric functions in common calculators expect radians unless the software is explicitly set to degrees.

Worked geometry example

Suppose a standard gear has mₙ = 3 mm, β = 30°, αₙ = 20°, and z = 20 teeth. Since cos 30° is about 0.8660, the transverse module is about 3.464 mm. The reference diameter is d = z mₜ, or about 69.28 mm. The transformed pressure angle is αₜ = arctan(tan 20° / cos 30°), approximately 22.80°. These results align with the normal-system example published by KHK for a 3 mm module and 30° helix angle.

Pitch is also plane-specific. Normal pitch is pₙ = πmₙ, while transverse pitch is pₜ = πmₜ. Do not multiply a normal module by π and call it transverse pitch, or use the normal module directly in a transverse reference-diameter calculation. In a mating external pair, the gears must use compatible modules and pressure angles in the same defined system; the helix angles have equal magnitude and opposite hand for parallel shafts.

What this table does not calculate

These formulas establish basic reference geometry only. They do not account for profile shift, working center distance, backlash, tooth thickness, undercut, contact ratio, accuracy class, root strength, pitting resistance, lubrication, or thermal expansion. A design that uses profile shift or nonstandard proportions needs the corresponding equations and boundary conditions rather than the unshifted reference diameter formula alone.

For load capacity, consult the applicable gear-rating standard and the detailed manufacturer data for the material, heat treatment, quality grade, face width, speed, and service factor. Confirm the actual mating gear, shaft arrangement, bearing stiffness, and housing alignment. Manufacturing errors and deflection can change tooth contact even when the nominal dimensions are correct.

When checking a drawing, label every dimension as normal or transverse. Record the normal module, normal pressure angle, helix angle, tooth count, hand, profile shift, and reference diameter. Verify the relationship d = z mₜ for an unshifted standard gear and check that the paired gear uses compatible values. If a catalog or design standard defines a different system or special case, use that controlling definition.

Source: KHK Gears, Calculation of Gear Dimensions, section 4.3, which explains normal and transverse helical-gear systems and includes calculation examples. This reference is limited to basic geometry; it is not a tooth-strength or gear-rating calculation.

Before calculating, identify whether the drawing starts with normal or transverse module. A module marked only “3” is ambiguous for a helical gear unless the system is stated. Record β with its direction and hand convention, and ensure that both mating gears use matching conventions. Keep intermediate values at full precision and round only the dimensions that will appear on the drawing.