Gears: module, formulas and ratios

November 3, 2026  ·  Indumor

A module 3 gear with 40 teeth measures 126 mm over the tips, not 120. The extra 6 mm are one addendum per side, and the reverse sum (outside diameter over the number of teeth plus two) settles most of the questions that reach the lathe with a broken gear in hand. How the teeth are milled is in chapter 13; how the blank is indexed, in chapter 12.

Module, diametral pitch and circular pitch

The module is the pitch diameter divided by the number of teeth, in millimeters. Two gears mesh only if they share the module and the pressure angle; the tooth counts can be anything. ISO 54 sorts the modules into two series, series 1 preferred and series 2 for when series 1 will not do.

Series 1 is preferred; series 2 is for when series 1 will not do. DIN 780 adds modules below 1 (0.5, 0.6, 0.7, 0.8, 0.9) for instruments and clockwork.

The circular pitch p is the distance from one tooth to the next along the pitch circle, π times the module. Machines of British or American origin define the tooth by diametral pitch (DP) instead, teeth per inch of pitch diameter, so a larger DP means a smaller tooth. Converting is dividing 25.4 by the number. A 10 DP gear is module 2.54, close to 2.5 but not equal, and a DP gear will not run properly with a module gear.

Spur gear formulas

Every dimension comes from the module. The ISO 53 and DIN 867 basic rack sets the addendum at one module, the dedendum at 1.25 modules and the pressure angle at 20°. The old 14.5° gears, narrower at the root and wider at the tip, survive in farm machinery and older machine tools, and they do not mesh with 20° gears.

To identify a gear with no drawing, count the teeth, measure the outside diameter and divide by z + 2. If it comes out as a module in the table, done; if it comes out odd, try DP = (z + 2) × 25.4 / d_a. The outside diameter stays reliable on worn flanks, since the tip does no work. What misleads is a profile-shifted pinion, which measures more than nominal.

Measuring the tooth with a gear tooth caliper

The gear tooth vernier caliper is a caliper with two verniers. The horizontal one measures the tooth thickness and the vertical one sets how far down from the tip the jaws sit. Since the real thickness is an arc of the pitch circle and a caliper can only measure a chord, two corrected values are used: the chordal thickness s_c, which you read, and the chordal addendum h_c, which you set.

Two details spoil the reading. h_c is counted from the tip, so a gear turned to an outside diameter other than nominal needs half the difference added or subtracted. And the table is for unshifted teeth; with profile shift the thickness is recalculated per DIN 3960. For a helical gear, enter with m_n and the virtual tooth number z_v = z / cos³β.

Few teeth: undercut and profile shift

Below 17 teeth at 20°, the cutter or the hob digs into the root of the tooth. That is undercut, which weakens the root and shortens the line of action. At 14.5° the limit rises to 32. Tolerating a slight undercut takes it down to 14 teeth at 20°.

To go lower there is profile shift, x: the tool is moved x·m away from the center, the tooth gets thicker at the root and the outside diameter grows by 2·x·m. A 12-tooth pinion at 20° needs x = 0.30 at least. Module and tooth count do not change; the center distance does (DIN 3960 has the calculation).

The practical value accepts a slight undercut that does not compromise the root; below it the profile has to be shifted.

Helical gears

In a helical gear the tooth leans at a helix angle β to the axis, and there are two modules. The normal module m_n, measured perpendicular to the tooth, is the cutter's; the transverse module m_t, in the plane of the gear, is the one that gives the diameter. The usual helix angles run from 15° to 30°: below 10° the smoothness is lost, and above 30° the axial thrust punishes the bearings. For the cutter and for chordal measurement the virtual number of teeth z_v is used, always larger than z.

The usual range is 15° to 30°. z_v is the virtual number of teeth (it selects the gear cutter and enters the chordal table); p_x is the axial pitch.

A parallel-shaft pair carries helices of opposite hand and the same β; if the shafts cross at 90°, both gears are of the same hand and the two angles add up to 90°. Put the hand on the order: a left-hand gear cut right-hand is scrap.

Worm and worm wheel

A worm is a trapezoidal thread meshing with a throated helical wheel. Besides the module (axial on the worm, transverse on the wheel), it is defined by the diametral quotient q = d₁ / m, the worm pitch diameter in modules, usually between 8 and 14. A small q gives a more efficient but more flexible worm; a large q, a stiff worm with lower efficiency. The lead angle comes from tan γ = z₁ / q, with z₁ the number of starts, 1 to 4. A single start at 4 or 5 degrees runs at 60 to 70 % efficiency; four starts at 25° reach 90 %.

μ of 0.03 to 0.05: hardened and ground worm on bronze, well lubricated; 0.08: start-up, poor lubrication or a cast iron wheel. Bearings and oil churning not included.

A single-start worm with a low γ is self-locking: the wheel cannot drive the worm, which is what hoists and lifting tables rely on. Self-locking is not braking; under vibration the load creeps, so anything that hangs gets a brake. The material pair is almost always a hardened and ground steel worm against a tin bronze wheel, because the contact is pure sliding.

Materials and hardness

A shop gear is cut from normalized 1045 bar and does fine on a slow machine. With more torque and speed it moves to quenched-and-tempered 4140, and with heavy loads to a carburizing steel such as 8620, cut soft, carburized, hardened and ground. The hardness values in the table are indicative; heat treatments are in chapter 20 and the hardness scales in chapter 21.

Ordering a gear takes eight pieces of data: tooth count, module (or DP), pressure angle, helix angle with its hand, face width, bore with its keyway, material and heat treatment. If the old part exists, take it along: ten minutes with the caliper and the chordal table settle more than a half-finished drawing. And if it meshes with a gear that stays in the machine, take that one too.

Frequently asked questions

How do you find the module of a gear?

Count the teeth and measure the outside diameter: m = d_a / (z + 2). A 40-tooth gear measuring 126 mm is module 3. If the result is not a standard value, try diametral pitch: DP = (z + 2) × 25.4 / d_a.

What is the difference between module and diametral pitch?

Two ways of stating tooth size. The module is millimeters of pitch diameter per tooth; DP is teeth per inch of pitch diameter. m = 25.4 / DP. An 8 DP gear is module 3.175 and does not mesh with module 3.

What is the minimum number of teeth on a gear?

At a 20° pressure angle, 17 teeth without undercut (14 if a slight undercut is tolerated); at 14.5°, 32. Below that, the profile is shifted by moving the tool outward (positive x).

Sources: ISO 53:1998 (basic rack); ISO 54:1996 (modules); ISO 1122-1 (gear vocabulary); DIN 780 (module series); DIN 867 (basic rack profile); DIN 3960 (definitions and parameters of involute cylindrical gears); DIN 3975 (worm gears, terms and definitions); ANSI/AGMA 1012 (nomenclature); ANSI/AGMA 2001-D04. Chordal thickness and chordal addendum computed with s_c = m·z·sin(90°/z) and h_c = m·[1 + (z/2)·(1 − cos(90°/z))]; worm efficiency with η = tan γ / tan(γ + ρ).