Gear Module and Diametral Pitch Calculator

Module, diametral pitch and circular pitch converted exactly in any direction, with the ISO 54 module series, the standard diametral pitch series, the tooth proportions each implies, and the near-miss table that shows why a 2 mm module gear and a 12 DP gear do not mesh.

Last updated 24 Sep 2026

Module, DP and circular pitch

Any one of the three → the other two, exactly

Which one do you have?The other two are computed and locked. DP = 25.4/m exactly, and the circular pitch is πm — so any one of the three fixes the other two.

Module m (mm)Tooth size in the metric system: the pitch diameter of a z-tooth gear is m·z. ISO 54 series 1 runs 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40, 50.

Diametral pitch P_d (teeth per inch of pitch diameter)Tooth size in the inch system, and note that it works BACKWARDS from module — a bigger DP is a smaller tooth. A 1 DP gear has one tooth per inch of pitch diameter, so its teeth are 25.4 mm module.

Circular pitch p (mm)The arc distance from one tooth to the next, measured along the pitch circle: p = πm. Occasionally used directly — CP racks and pinions are sold on it because a whole-number circular pitch makes a rack easy to butt end to end.

Tooth count, to see what this costs in pitch diameterUsed only to turn the tooth-size mismatch into a diameter you can measure. The error in pitch diameter is the error in module multiplied by the tooth count, which is why a near-miss that looks negligible per tooth is not negligible on a blank.

Not a circuit: two ladders of standard tooth sizes on one logarithmic axis, from half a millimetre of module to 120. Above the line, the ISO 54 module series — long ticks for series 1, short ones for series 2. Below the line, the standard diametral pitch series, each tick placed at the module that pitch really is. The whole point of the figure is that the two sets of ticks NEVER COINCIDE: not one standard module has a standard diametral pitch, and not one standard pitch has a standard module. Look at the near misses, which are the expensive ones — 12 DP sits just to the right of module 2, because it is 2.1167 mm, and a pair of gears made to those two numbers will engage and then wear out. The arrow above the axis is your own tooth size, quantised to about a fortieth of the figure.

2.00000mmExample

A 2 mm module, and a 40-tooth gear to put it on

One exact conversion and one exact proportion

P_d = 25.4 / m · m = 25.4 / P_d · p = πm · p = 25.4π / P_d · d = m·z = z / P_d (inches)

m: module, millimetres of pitch diameter per tooth. A size: bigger m, bigger tooth
P_d: diametral pitch, teeth per inch of pitch diameter. A density: bigger P_d, SMALLER tooth. This inversion is the source of most of the confusion
p: circular pitch, the arc from one tooth to the next along the pitch circle
25.4: millimetres per inch, exactly, by definition since 1959. So the module/DP conversion is exact and the near misses in the table below are exact too — they are not rounding, they are real differences in tooth size

Worked example

A 2 mm module, and a 40-tooth gear to put it on
The two conversions are one line each and both are exact. P_d = 25.4/m = 25.4/2 = 12.7000 DP, and the circular pitch is πm = 6.28319 mm. Nothing on this page is harder than that, which is exactly why the rest of it exists
Module 2 is an ISO 54 series-1 value, so it is a size you can buy a hob, a shaper cutter, a master gear and stock blanks for. Its diametral pitch, 12.7, is NOT a standard pitch — and that is true of every module in series 1. Not one of the eighteen has a whole-number DP
So consider the near miss everybody makes. The nearest standard pitch to 12.7 is 12 DP, and 12 DP is 25.4/12 = 2.11667 mm module. That is 5.83 per cent bigger teeth. It is not a tolerance and it is not a rounding: the two gears have genuinely different tooth sizes and cannot mesh
Turn that into something measurable, because a percentage per tooth sounds ignorable. On a 40-tooth gear the pitch diameter is m·z: 80.0 mm at module 2, and 84.6667 mm at 12 DP. That is 4.667 mm of diameter difference on one gear, and it grows with the tooth count. Two such gears run at a centre distance neither of them was cut for
The tooth proportions at module 2, which is the other half of what choosing a module means: addendum 2.0 mm, dedendum 2.50 mm, whole depth 4.50 mm, tooth thickness on the pitch circle 3.1416 mm, root fillet 0.76 mm. Every one of those scales linearly with the module, so the module is the tooth, in millimetres
One last number worth knowing. Module and diametral pitch cross at √25.4 = 5.0398: a gear of about 5.04 mm module is also about 5.04 DP. It is the only size at which the two numbers agree, it is not a standard value in either system, and it is a useful landmark for remembering which side of the inversion you are on — below it the DP number is the bigger of the two, above it the module is

ISO 54 modules, and the diametral pitch each one really is

ISO 54:1996 table 1. Preference goes to series 1; the standard says in as many words that “the module 6,5 of series II should be avoided”, which is why it is parenthesised here. Now look down the second and fourth columns, because this is the whole point of the page: NOT ONE standard module has a whole-number diametral pitch. The closest any of them comes is module 2, which is 12.7 DP — and 12.7 is not a standard pitch either. The two systems were never meant to coexist: ISO 54:1977 did carry a diametral pitch table, but it introduced it “only on a provisional basis” and said the values “will be deleted after the period necessary to allow conversion to the metric system”. They were. ISO 54:1996 has no diametral pitch table at all. These dimensions come from a published standard’s table, not from a formula. The standard itself is cited below and the printed values are attributed to the catalogue they were taken from; a different publisher may round differently in the last digit.
Series 1 (preferred, mm)	… as DP	Series 2 (mm)	… as DP
1.000	25.4000	1.125	22.5778
1.250	20.3200	1.375	18.4727
1.500	16.9333	1.750	14.5143
2.000	12.7000	2.250	11.2889
2.500	10.1600	2.750	9.2364
3.000	8.4667	3.500	7.2571
4.000	6.3500	4.500	5.6444
5.000	5.0800	5.500	4.6182
6.000	4.2333	(6.5) — to be avoided	3.9077
8.000	3.1750	7.000	3.6286
10.000	2.5400	9.000	2.8222
12.000	2.1167	11.000	2.3091
16.000	1.5875	14.000	1.8143
20.000	1.2700	18.000	1.4111
25.000	1.0160	22.000	1.1545
32.000	0.7937	28.000	0.9071
40.000	0.6350	36.000	0.7056
50.000	0.5080	45.000	0.5644

The near misses: every common module against the closest standard diametral pitch

This is the table people come here for, and the last two columns are the ones that matter. Take the most quoted near-miss: a 2 mm module gear and a 12 DP gear. 12 DP is 25.4/12 = 2.1167 mm module, so its teeth are 5.83 per cent bigger — and on a 40-tooth blank that is 4.67 mm of pitch diameter. They do not mesh, they cannot be made to mesh, and the reason they get confused is that the two numbers look close and the gears look identical. Only ONE pair in this table is closer than one per cent. Where the error is under about half a per cent a pair will often run — badly, with the contact shifted off the pitch point, a wrong contact ratio and accelerated wear — which is worse than an outright mismatch because it ships. The same designation can mean different dimensions in different standards families — ANSI against ISO, inch against metric, one national standard against another. The family used here is named beside every figure; check which one your part was made to.
Module (mm)	Its exact DP	Nearest standard DP	That DP’s module (mm)	Tooth-size error	Pitch-diameter error on a 40-tooth gear (mm)
1.00	25.4000	24.000	1.05833	+5.83%	+2.333
1.25	20.3200	20.000	1.27000	+1.60%	+0.800
1.50	16.9333	16.000	1.58750	+5.83%	+3.500
2.00	12.7000	12.000	2.11667	+5.83%	+4.667
2.50	10.1600	10.000	2.54000	+1.60%	+1.600
3.00	8.4667	8.000	3.17500	+5.83%	+7.000
4.00	6.3500	6.000	4.23333	+5.83%	+9.333
5.00	5.0800	5.000	5.08000	+1.60%	+3.200
6.00	4.2333	4.000	6.35000	+5.83%	+14.000
8.00	3.1750	3.000	8.46667	+5.83%	+18.667
10.00	2.5400	2.500	10.16000	+1.60%	+6.400
12.00	2.1167	2.000	12.70000	+5.83%	+28.000

What the module actually buys you, tooth by tooth

Every column is simply proportional to the module, which is the reason module is a more useful number than diametral pitch: it IS the tooth size, in millimetres, and you can read the tooth off it. Doubling the module doubles the tooth thickness, doubles the depth, doubles the fillet and — at a fixed tooth count — doubles the diameter. What it does to strength is better than proportional in one sense and worse in another: bending stress falls inversely with module at a fixed tangential load, which the bending stress page computes, but a coarser tooth on the same diameter means fewer teeth and a lower contact ratio, so the load is shared by less of the mesh. Choosing a module is choosing that trade. These dimensions come from a published standard’s table, not from a formula. The standard itself is cited below and the printed values are attributed to the catalogue they were taken from; a different publisher may round differently in the last digit.
Module (mm)	DP	Circular pitch (mm)	Addendum (mm)	Dedendum (mm)	Whole depth (mm)	Tooth thickness at the pitch circle (mm)	Root fillet (mm)
0.50	50.800	1.571	0.500	0.625	1.125	0.785	0.190
1.00	25.400	3.142	1.000	1.250	2.250	1.571	0.380
1.50	16.933	4.712	1.500	1.875	3.375	2.356	0.570
2.00	12.700	6.283	2.000	2.500	4.500	3.142	0.760
3.00	8.467	9.425	3.000	3.750	6.750	4.712	1.140
4.00	6.350	12.566	4.000	5.000	9.000	6.283	1.520
6.00	4.233	18.850	6.000	7.500	13.500	9.425	2.280
10.00	2.540	31.416	10.000	12.500	22.500	15.708	3.800

One exact conversion, and why no standard module has a standard diametral pitch

The conversion is one line and it is exact. P_d = 25.4/m, because there are exactly 25.4 millimetres in an inch by definition, and the circular pitch is πm. So module 2 is 12.7 DP, module 1 is 25.4 DP, and 8 DP is 3.175 mm module. If that were all anyone needed, this page would not need to exist. What people actually need is the thing the conversion hides: the two systems’ standard sizes do not line up anywhere.

Not one ISO 54 module has a whole-number diametral pitch. Series 1 is 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40, 50 mm, and their diametral pitches are 25.4, 20.32, 16.933, 12.7, 10.16, 8.4667, 6.35, 5.08, 4.2333, 3.175… — not a standard value among them. The same is true in reverse. This is not an accident of rounding: ISO 54:1977 DID carry a diametral pitch table, and introduced it “only on a provisional basis” with the note that the values “will be deleted after the period necessary to allow conversion to the metric system”. They were deleted. ISO 54:1996 has no diametral pitch table at all.

A 2 mm module gear and a 12 DP gear do not mesh. This is the specific confusion the page is built around, because 12 is the nearest standard pitch to module 2’s 12.7 and the two numbers look close. 12 DP is 25.4/12 = 2.11667 mm module, so its teeth are 5.83 per cent larger. On a 40-tooth gear that is 4.67 mm of pitch diameter — not a tolerance, a different gear. The near-miss table below does this for every common module, and the column worth reading is the last one, because a percentage per tooth sounds ignorable and a diameter does not. Watch especially for mismatches under about one per cent: those pairs will physically engage, and a pair that nearly meshes is far more expensive than one that obviously cannot, because it ships.

What choosing a module actually decides. Everything about the tooth, linearly: addendum 1 m, dedendum 1.25 m, whole depth 2.25 m, thickness at the pitch circle πm/2, root fillet 0.38 m, all from ISO 53’s standard basic rack. Double the module and you double all of it. That makes the trade explicit: a coarser tooth is stronger in bending at the same load, which the bending stress page quantifies, but at a fixed diameter it means fewer teeth and a lower contact ratio, so less of the mesh shares the load and the pair is noisier. It also decides your centre distance, because that is m(z₁ + z₂)/2 with no freedom in it.

Where the other conversions live. This page converts tooth sizes and nothing else. General unit conversion — thread pitch and TPI, drill sizes, wire and sheet gauges, surface roughness, hardness scales, viscosity — belongs to the converters section of this site rather than to the mechanical calculators, and is not linked from here because these pages only link within the mechanical set. If you are converting a thread pitch and not a gear tooth, that is the one you want.

Frequently asked questions

How do I convert module to diametral pitch?

P_d = 25.4/m, and m = 25.4/P_d. The conversion is exact, because the inch has been exactly 25.4 mm by definition since 1959. Module 2 is 12.7 DP; 10 DP is 2.54 mm module; module 1 is 25.4 DP. The one thing to keep straight is that the relationship is inverse: a LARGER diametral pitch is a SMALLER tooth, because it counts teeth per inch of diameter rather than millimetres of diameter per tooth.

Will a 2 mm module gear mesh with a 12 DP gear?

No. 12 DP is 25.4/12 = 2.11667 mm module, which is 5.83 per cent coarser than 2 mm. On a 40-tooth gear that is 4.67 mm of pitch diameter. They will not run together at any centre distance. The confusion is understandable — 12 is the nearest standard diametral pitch to module 2’s exact 12.7 — and it is worth knowing that no standard module has a standard diametral pitch, so this kind of near miss is the normal case rather than the exception.

What are the standard gear modules?

ISO 54 series 1, which is the preferred series: 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40 and 50 mm. Series 2 fills the gaps: 1.125, 1.375, 1.75, 2.25, 2.75, 3.5, 4.5, 5.5, (6.5), 7, 9, 11, 14, 18, 22, 28, 36 and 45. The standard says to prefer series 1 and, unusually, singles out one value to avoid — “the module 6,5 of series II should be avoided”. Below module 1 there is no ISO series, and fine-pitch sizes are set by the stock-gear catalogues.

What is circular pitch and when is it used?

The arc distance from one tooth to the next, measured along the pitch circle: p = πm. It is mostly an intermediate quantity, but it is used directly for CP racks and pinions, where a whole-number circular pitch lets lengths of rack be butted end to end without the joint landing in the middle of a tooth space. A whole-number circular pitch is almost never a standard module: a 10 mm CP rack is 3.1831 mm module.

Is a bigger diametral pitch a bigger tooth?

No — the opposite, and this catches people constantly. Diametral pitch counts teeth per inch of pitch diameter, so a 48 DP gear has very fine teeth (0.529 mm module) and a 1 DP gear has enormous ones (25.4 mm module). Module runs the intuitive way round: it is millimetres of pitch diameter per tooth, so it IS the tooth size. The two numbers cross at √25.4 = 5.0398, which is the only size at which a gear’s module and its diametral pitch are the same number.

Can I cut a gear to a non-standard module?

Yes, and it is done — usually when a fixed centre distance and a fixed ratio have to be satisfied at the same time. The cost is tooling: hobs, shaper cutters, master gears, gear-tooth micrometer settings and stock blanks are all made to the series, so a non-standard module means either a special cutter or generating the tooth some other way. This page shows the nearest series-1 and series-2 modules with the percentage difference and what that difference costs in pitch diameter at your tooth count; if the tooth counts can absorb it, moving to a standard module is almost always cheaper.

Related calculators

References

ISO 54:1996, Cylindrical gears for general engineering and for heavy engineering — Modules, and ISO 54:1977, the earlier edition. The module series here is ISO 54’s table 1, read from the freely published previews of both editions and in both English and French. Three readings were needed: the 1996 English preview truncated series I at 32 and rendered the parenthesised series II value as 6.75, the French preview truncated series II at 5.5 and rendered the avoided module as 63, and the 1977 edition — which carries the full 1 to 50 mm range in one table — settled both as 6.5. The 1977 edition is also the only one with a diametral pitch table, which it introduced “only on a provisional basis” and said “will be deleted after the period necessary to allow conversion to the metric system”. It was: ISO 54:1996 has no diametral pitch table at all.
ISO 53:1998, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile. Cited by number. What this page takes from it is four coefficients, not a table: addendum h_aP = 1 m, bottom clearance c_P = 0.25 m, hence dedendum h_fP = 1.25 m and whole depth 2.25 m, with a root fillet ρ_fP = 0.38 m and a 20° profile angle. The standard specifies exactly ONE pair of values for c_P and ρ_fP, which is why they are constants here and not a choice. Confirmed independently against the DIN 867 basic rack, which is the same profile.
Stock Drive Products / Sterling Instrument, Elements of Metric Gear Technology (the technical section of catalogue D805). The source for two things taken as printed here: the minimum tooth count free of undercut — “for 14.5° the value of z_c is 32, and for 20° it is 18” — and the contact ratio guidance, “it is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1, calculated for all tolerances at their worst case values.” Also the source for the self-locking condition in the form (cos 20° sin γ − μ cos γ) ≤ 0, which is what puts the normal pressure angle into the threshold.
KHK (Kohara Gear Industry), Gear Technical Reference — the chapters on gear backlash, mounting accuracy and the surface durability of worm gears. Consulted for the backlash vocabulary (circumferential j_t, normal j_n, radial j_r, angular) and for the way a stock-gear maker states a backlash it will actually hold.
Drivetrain Hub, Gear Geometry notebooks, chapters 2 and 3 (spur and helical gears). The source that states the undercut limit as 17.097 rather than as 17, and for the helical relations m_t = m_n/cos β, tan α_t = tan α_n/cos β and the base helix angle. Its printed contact-ratio expression uses the ROOT diameters where the derivation needs the BASE diameters, so the contact ratio on this site is derived from the line of action and checked against a direct geometric construction instead.