Module (m) and diametral pitch (DP) describe the same physical quantity, gear tooth size, but use opposite ratios in opposite units: m = pitch diameter (mm) / number of teeth, while DP = number of teeth / pitch diameter (inches) [S2][S3]. The two are exact reciprocals scaled by 25.4 mm/in, so m (mm) = 25.4 / DP and DP (1/in) = 25.4 / m [S3][S4].
The unit choice carries a physical inversion: a higher DP value means a smaller, finer tooth (more teeth per inch of pitch diameter), whereas a higher module value means a larger, coarser tooth (each tooth occupies a bigger slice of the pitch circle) [S3][S6]. ISO governs the metric module system worldwide, while AGMA inch-system standards govern diametral pitch across most of North America [S3].
Formulas, Units, and the 25.4 Conversion Constant
Both systems anchor on pitch diameter d, but the algebra flips: for a 25-tooth gear of module 3, d = m × z = 3 × 25 = 75 mm, while the diametral-pitch equivalent works as DP = z / d(in) so that the same physical size in inches would invert the calculation [S2][S8]. The single conversion identity, m (mm) = 25.4 / DP, is reproduced identically across the engineering references and lets a designer switch systems without recomputing the underlying geometry [S3][S4][S7].
Practical worked examples from the conversion table: a module 1 gear equals DP 25.4, a module 2 gear equals DP 12.70, a module 4 gear equals DP 6.35, and a module 10 gear equals DP 2.54 [S2][S3]. A common coarse-imperial DP 20 gear (0.05 in addendum) corresponds to module 1.27, while a DP 32 gear (0.0313 in addendum) corresponds to module 0.79375 [S2]. For rack and pinion drives, the same formulas apply, but the linear pitch becomes the deciding parameter and module is the preferred European metric term [S7].
Standard Stock Pitches: Module Series vs DP Series
ISO 53 and equivalent national standards define a preferred series of modules, with 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, and 10 mm being the most commonly stocked coarse-pitch metric sizes, and 0.5, 0.8, 1.0 being common fine-pitch metric sizes [S2][S3]. The AGMA standard stock series for diametral pitch includes integer values 20, 24, 32, 48, 64, 72, 80, 96, 120, and 200, each with its own addendum, dedendum, whole depth, and circular pitch numbers locked by the standard [S2].
The two series are not numerically equal, they interleave. For example, module 1 (DP 25.4) is not the same as DP 25; module 2 (DP 12.70) is not the same as DP 12; module 4 (DP 6.35) is not the same as DP 6 [S3]. This is the most common specification error when substituting parts across standards: a metric module 2 gear has a tooth 0.70 mm finer in module terms than a DP 12 gear and is not interchangeable without recalculating center distance [S2][S7].
Decision Criteria: When to Specify Module vs Diametral Pitch
Specify module (m, mm) when the driven machine is built to ISO metric standards, when mating components (pinions, racks, gear pumps, gear reducers) are sourced from European or Asian supply chains, and when the mating gear or gear coupling carries a metric part number. Specify diametral pitch (DP, 1/in) when the drivetrain is built to AGMA imperial standards, when the mating gear, shaft, or housing bore is dimensioned in inches, and when replacement parts must match a legacy North American machine tag. [S3]
For mixed-fleet or imported-equipment situations, the conversion table becomes the specification bridge. A helical gear reducer or industrial gear sourced from a metric-standard factory will carry a module stamp (e.g. m3, m4), and the equivalent DP value is read directly as 25.4 / m: m3 equals DP 8.4667, m4 equals DP 6.35, m6 equals DP 4.2333 [S2][S3]. Conversely, a DP 20 gear from a US parts list converts to module 1.27, not to the rounded module 1.25 that an unwary buyer might pick off a metric shelf [S2][S7].
Comparison: Module vs Diametral Pitch on Spec-Level Criteria
On tooth-count-per-pitch-diameter: DP = teeth per inch, higher value = smaller tooth; m = mm per tooth, higher value = larger tooth. The numerical direction of "bigger" is inverted between the two systems, so a DP 32 part and a module 0.8 part (close to DP 31.75) are equivalent physical sizes, but their labels move in opposite directions as the tooth gets finer [S3][S6].
On addendum/dedendum/whole depth, the AGMA standard stock table for DP 20 specifies addendum 0.0500 in, dedendum 0.0579 in, whole depth 0.1079 in, and circular pitch 0.1571 in, while a module 4 gear (close to DP 6.35) carries addendum 4 mm, dedendum 4.63 mm, whole depth 8.63 mm by the ISO coarse-pitch rules [S2]. On unit system, module maps cleanly to mm-based drawings and to linear module sizing where rack pitch and pinion module share a single number, while DP maps cleanly to inch-based shaft and bore fits and to US-standard gear-cutting hobs [S3][S7].
On interchangeability risk, the highest-risk path is converting a DP value to the nearest preferred module via mental rounding (DP 20 to module 1.25 instead of the exact module 1.27) and then specifying the gear by the rounded number, which shifts center distance by 0.02 mm per tooth pair and can break mesh contact [S2][S4]. The safe procedure is to keep the exact 25.4 / DP result and only round to the nearest ISO preferred module when a small center-distance change is acceptable and the mating gear is re-cut in the same module.
Common Substitution Errors and Measurement Catches
Three failure modes recur in field reports and on machinist forums. First, the inverse-direction trap: an engineer used to DP will read "module 3" and think it is coarser than "module 1," but module 3 is in fact 3 mm of pitch diameter per tooth, three times larger than module 1, the opposite intuition of the DP world [S3][S6]. Second, the DP 25 vs module 1 confusion: DP 25 is not module 1, it is module 25.4 / 25 = 1.016, and using a module 1 part in place of a DP 25 part shifts pitch diameter by 1.6 percent [S2][S4].
Third, circular-pitch substitution: circular pitch CP (in or mm) is the arc length along the pitch circle between corresponding points on adjacent teeth, equal to π × m in metric and π / DP in inches, and is sometimes quoted on legacy drawings in place of module or DP, which makes the unit-of-length question (in vs mm) the deciding factor rather than the ratio itself [S2][S3]. For rack and pinion drives, the same three failure modes apply but with linear pitch replacing the angular pitch, and module remains the preferred metric designation per [S7].
Use Cases: Gear Pumps, Reducers, and Rack-and-Pinion Drives
External gear pumps almost always carry an exact module or DP stamp on the gear set, because the displacement per revolution is a direct function of tooth size, and substitution of a metric-module gear into a DP-rated pump body changes the flow rate by the same 1.6 percent-or-larger error noted above [S4][S7]. For multi-stage gear reducers, each stage carries its own module or DP, and mixing standards within one reducer is not done in practice because the gears within a stage must share module, pressure angle, and helix angle to mesh.
For rack-and-pinion linear drives, module is the more common modern spec because the linear rack pitch (π × m, in mm) maps directly to the linear travel per pinion tooth, while DP-based racks are still common on US-built machine tools and AGMA-class packaging equipment [S7]. When ordering a replacement rack for an imported metric machine, read the pinion's module stamp first, divide 25.4 by it to get the DP equivalent, and then verify the rack's linear pitch in mm matches π × m rather than π / DP, which is the most common rack-vs-pinion mismatch in field replacements [S2][S7].
For replacement gearing, a working 2026 procedure: identify the tooth-count (z), measure the pitch diameter in mm if possible, compute m = d / z, and round to the nearest preferred ISO module only if the center distance can be shimmed, otherwise specify the exact m value and have the gear cut to it [S2][S3][S4]. The next data point to watch is any AGMA or ISO update to the preferred series tables, since the standard preferred modules (1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10) and standard stock DPs (20, 24, 32, 48, 64, 72, 80, 96) are the de-facto cross-reference points that drive global supply of replacement gears and racks [S2][S3].
This topic is covered further in Standardized Mold Base vs. Interchangeable Cavity Inserts: Decision Logic.