Gear module vs diametral pitch

Every spur or helical gear has a tooth size, and there are two standard ways to express it. The metric world uses the module; the imperial world uses diametral pitch. They measure exactly the same physical thing — how big each tooth is — but they run in opposite directions, which is the single biggest source of confusion. Get the unit system straight and the rest of the gear geometry falls out of a handful of simple formulas. You can check any of them with the gear module calculator.

Module — the metric tooth-size unit

The module (symbol m) is the pitch diameter divided by the number of teeth, measured in millimetres:

m = d / z

where d is the pitch diameter in mm and z is the tooth count. Module is an SI unit of length, so it has a tidy physical meaning: it is the amount of pitch diameter contributed by each tooth. A bigger module means bigger teeth. A module-1 gear has small, fine teeth; a module-6 gear has large, coarse teeth that can carry far more load. Common standard modules are 0.5, 1, 1.5, 2, 2.5, 3, 4, 5 and 6 mm.

Diametral pitch — the imperial tooth-size unit

Diametral pitch (symbol DP or P) is the number of teeth per inch of pitch diameter:

DP = z / d  (d in inches)

This is the inverse of the idea behind module. Because DP counts teeth crammed into a fixed inch of diameter, a bigger DP means smaller teeth — the exact opposite of module. A DP-4 gear has big, coarse teeth; a DP-48 gear has tiny, fine teeth like those in a clock or a small instrument drive. Whenever you switch between the two systems, remind yourself that they run in opposite directions: large module and large DP are at opposite ends of the size scale.

Converting between module and DP

Because they describe the same tooth size in different units, the conversion is just the inch-to-millimetre factor of 25.4:

m = 25.4 / DP    DP = 25.4 / m

The product of module and DP is always 25.4. So a module-2 gear corresponds to DP = 25.4 / 2 = 12.7, and a DP-10 gear corresponds to m = 25.4 / 10 = 2.54 mm. Notice that round metric modules almost never map to round DP values, and vice versa — which is exactly why metric and imperial gears do not interchange in practice even though the maths connects them.

Conversion table

Module m (mm)	Diametral pitch DP = 25.4 / m
0.5	50.8
1	25.4
1.5	16.93
2	12.7
2.5	10.16
3	8.47

Read it the other way for common imperial sizes: DP 48 is module 0.529, DP 32 is module 0.794, DP 24 is module 1.058, and DP 20 is module 1.27. The fact that none of these are round metric numbers underlines the point — the two standards were never meant to line up.

Everything else follows from the module

Once you know the module and the tooth count, the rest of the standard tooth geometry is fixed. For a full-depth involute tooth (the usual case), the relations are:

• Pitch diameter: d = m · z
• Circular pitch (arc length from tooth to tooth along the pitch circle): p = π · m
• Addendum (height of the tooth above the pitch circle): a = m
• Dedendum (depth below the pitch circle): b = 1.25 · m
• Whole depth: h = 2.25 · m
• Outside (tip) diameter: Do = m · (z + 2)

The outside diameter picks up the extra 2m because the addendum (one module) sticks out on both sides of the pitch circle. If you work in DP, the same relations hold with 1/DP in place of m and lengths in inches: addendum = 1/DP, whole depth = 2.25/DP, outside diameter = (z + 2) / DP, and so on. The involute gear geometry guide explains the base circle and tooth-thickness formulas, and the involute gear calculator will draw the actual tooth profile from these inputs and export it as DXF.

Mating gears must share module and pressure angle

For two gears to mesh smoothly, they must have the same module (or DP) and the same pressure angle — usually 20° on modern gearing, with 14.5° on older designs. The module match guarantees the teeth are the same size so they interlock; the pressure-angle match guarantees the tooth flanks have the same shape so they roll against each other without binding. The tooth counts can be completely different — that difference is precisely what creates the gear ratio — but the tooth size and flank angle have to agree. If either differs, the gears simply will not run together.

Worked example

Take a 20-tooth gear at module 2. The pitch diameter is:

d = m · z = 2 × 20 = 40 mm

The outside diameter is m(z + 2) = 2 × 22 = 44 mm, the circular pitch is π × 2 ≈ 6.28 mm, the addendum is 2 mm, the dedendum is 1.25 × 2 = 2.5 mm, and the whole depth is 2.25 × 2 = 4.5 mm. The imperial near-equivalent of the tooth size is:

DP = 25.4 / m = 25.4 / 2 = 12.7

There is no standard DP at 12.7, so you would not find an off-the-shelf imperial gear that meshes with this metric one — the nearest stock sizes are DP 12 (module 2.117) and DP 13 (module 1.954), both of which have subtly different teeth. To pair with this gear you stay in the module system and pick another module-2, 20°-pressure-angle gear of whatever tooth count gives the ratio you need. These are the same numbers the gear module calculator returns for these inputs.

Common mistakes

Three errors come up again and again:

• Mixing a module gear with a DP gear. They look interchangeable on paper, but standard module and standard DP values almost never coincide, so a "metric" and an "imperial" gear from different catalogues will usually not mesh. Match the actual tooth size, not the label.
• Confusing module with the gear ratio. Module is the tooth size and is the same on both gears in a pair; the ratio comes from the difference in tooth counts. Changing the module rescales the whole gear; changing the tooth count changes the ratio.
• Assuming DP is metric. Diametral pitch is an imperial unit — teeth per inch of diameter — and it gets larger as teeth get smaller. Treating a DP number as if it were a module in millimetres gives an answer off by a factor of roughly 25.

Keep those three straight and the two systems stop fighting you: module and DP are just two scales for one tooth size, linked by the constant 25.4, with every other gear dimension following from the module and the tooth count.

Frequently asked questions

How do I convert module to diametral pitch?

Use m = 25.4 / DP and DP = 25.4 / m. The 25.4 is millimetres per inch. For example, a module of 2 mm converts to DP = 25.4 / 2 = 12.7, and DP 10 converts to m = 25.4 / 10 = 2.54 mm. Note that the relationship is inverse: a bigger module means bigger teeth, but a bigger DP means smaller teeth.

Can a module gear mesh with a diametral-pitch gear?

Only if they happen to land on the same tooth size and pressure angle. Module and DP are just two unit systems for the same physical tooth size, so a module-2 gear and a DP-12.7 gear are identical and will mesh. But standard module sizes (1, 2, 3…) and standard DP sizes (8, 10, 12…) almost never coincide, so in practice you should not mix a metric-module gear with an imperial-DP gear off the shelf.

Is module the same as the gear ratio?

No. Module describes the size of an individual tooth and is shared by both gears in a mating pair. The gear ratio is the ratio of tooth counts (or pitch diameters) between the two gears and sets how speed and torque change through the mesh. Two gears with very different ratios can have the same module, and two gears with the same ratio can have different modules.

Last reviewed: 2026-05-29.