Sprocket Geometry Calculator

Every diameter a sprocket drawing needs, from the tooth count, the pitch and the roller — pitch, outside, root, maximum hub, and the two caliper diameters, because the even-tooth and odd-tooth figures are different numbers and measuring an odd sprocket with the even one is a classic rejection. ASME B29.1 forms throughout, with the chordal speed variation that explains the 17-tooth minimum, and an explicit list of what the standard contains that this page will not guess at.

Last updated 24 Sep 2026

Sprocket geometry

Teeth and pitch → every diameter

Number of teethThe whole page turns on this and the pitch. An odd count is not a mistake — it wears the chain more evenly — but it cannot be measured across the flats, which is what the two caliper figures below are about.

Chain sizeANSI and ISO use the same pitch and outside diameter formulas. What differs is the roller, and the roller sets the root and caliper diameters — ANSI 40 and ISO 08B-1 share a 12.70 mm pitch and differ by 0.59 mm on both.

Chain pitch (mm)

Roller diameter (mm)On ANSI 25 and 35 this is the bushing diameter: the numbering rule’s trailing 5 means the chain has no rollers, and the bushing is what seats in the tooth.

Not a circuit: one sprocket's four diameters as concentric circles, each drawn to scale against the pitch circle. The single radial line running out to the lower right crosses all of them, so the four crossings on it are the four diameters in the order they are listed on the right — outside, pitch, root, and on an ODD tooth count a fourth and slightly smaller one, the caliper diameter, which is what a vernier can actually reach across when no two tooth spaces are opposite each other. Watch that circle appear and disappear as you step the tooth count by one. On the left, the two radii bounding ONE tooth at the real 360°/N, with a chain roller seated on the pitch circle between them, also to scale — the roller is a much smaller part of a big sprocket than people draw it. The four circles are gated on four separate ratios, one each, so no part of this drawing depends on two moving quantities at once. The gap between the outside and pitch circles is the half-tooth that the PD + 0.6p shortcut overstates.

85.211mmExample

21 teeth for ANSI 40 chain — 12.70 mm pitch, 7.92 mm roller

Three numbers in, six diameters out

PD = p / sin(180°/N) · OD = p(0.6 + cot 180°/N) · BD = PD − D_r · caliper (even N) = BD · caliper (odd N) = PD·cos(90°/N) − D_r · MHD = p(cot 180°/N − 1) − 0.030 in

p: chain pitch. For an ANSI chain it is the leading digits of the number over eight, in inches, exactly
N: tooth count. Everything on the page is a function of it, and three separate things get worse as it falls: the chordal speed variation, the accuracy of the PD + 0.6p shortcut, and the room left for a bore
D_r: roller diameter — the bushing diameter on ANSI 25 and 35, which have no rollers. It appears in the root and caliper diameters and nowhere else
PD = p / sin(180°/N): the circle the roller CENTRES lie on. Exact: N chords of length p inscribed in a circle
OD = p(0.6 + cot 180°/N): B29.1’s approximate outside diameter, and ISO 606’s tip diameter dₐ. The word approximate is the standard’s. What is exact is that it is smaller than PD + 0.6p by p·tan(90°/N)
caliper, odd N: PD·cos(90°/N) is the chord between the two roller seats nearest to opposite, because on an odd sprocket no two are opposite. Confirmed here by construction: place the seats and measure the longest gap
chordal variation = 1/cos(180°/N) − 1: the chain’s speed ripple over one tooth. Derived, then confirmed by simulation. 1.73% at 17 teeth, 6.42% at nine

Worked example

21 teeth for ANSI 40 chain — 12.70 mm pitch, 7.92 mm roller
PD = 12.7 / sin(180°/21) = 12.7 / sin(8.5714°) = 12.7 / 0.149042 = 85.211 mm
OD = 12.7 × (0.6 + cot 8.5714°) = 12.7 × (0.6 + 6.63457) = 91.879 mm. The PD + 0.6p shortcut would have said 92.831 mm, which is 0.952 mm too big — exactly p·tan(90°/21)
Root diameter = 85.211 − 7.92 = 77.286 mm
21 is ODD, so the caliper diameter is not the root diameter. It is PD·cos(90°/21) − roller = 85.211 × 0.997204 − 7.92 = 77.048 mm — 0.24 mm under what a vernier across the flats of an even sprocket would read
Maximum hub and groove diameter = 12.7 × (cot 8.5714° − 1) − 0.762 mm = 70.797 mm
And the reason 21 teeth rather than 13: the chordal speed variation is 1/cos(180°/21) − 1 = 1.130%, against 2.99% on 13 teeth and 6.42% on nine

Every diameter, for ANSI 40 chain — 12.70 mm pitch, 7.92 mm roller

Change the pitch or the roller at the top of the page and the table above the chart recomputes for your chain; this one is fixed at ANSI 40 so there is something stable to check against. Four of the five diameter columns come straight from ASME B29.1 as AmesWeb print them: pitch p/sin(180°/N), outside p(0.6 + cot 180°/N), bottom PD − Dr, and maximum hub and groove p(cot 180°/N − 1) − 0.030 in. The caliper column switches form with the parity of the tooth count, which is the next table. Note what the maximum hub diameter does at low tooth counts: on nine teeth of chain 40 it is 21.4 mm, which after a bore and a keyway leaves very little sprocket. 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.
Teeth	Pitch Ø (mm)	Outside Ø (mm)	Root Ø (mm)	Caliper Ø (mm)	Which caliper form	Max hub Ø (mm)	Chordal variation (%)
9	37.132	42.513	29.208	28.643	odd	21.431	6.418
11	45.078	50.872	37.153	36.695	odd	29.790	4.222
13	53.068	59.146	45.143	44.756	odd	38.064	2.993
15	61.084	67.369	53.159	52.824	odd	46.287	2.234
17	69.116	75.559	61.191	60.896	odd	54.477	1.732
19	77.159	83.727	69.234	68.971	odd	62.645	1.383
21	85.211	91.879	77.286	77.048	odd	70.797	1.130
23	93.268	100.019	85.343	85.126	odd	78.937	0.940
25	101.330	108.151	93.405	93.205	odd	87.069	0.795
30	121.498	128.452	113.573	113.573	even	107.370	0.551
35	141.679	148.729	133.754	133.611	odd	127.647	0.404
40	161.868	168.989	153.943	153.943	even	147.907	0.309
45	182.062	189.238	174.137	174.026	odd	168.156	0.244
57	230.541	237.811	222.616	222.529	odd	216.729	0.152
76	307.320	314.678	299.395	299.395	even	293.596	0.085
95	384.111	391.521	376.186	376.134	odd	370.439	0.055
114	460.907	468.352	452.983	452.983	even	447.270	0.038

Why an odd-tooth sprocket cannot be measured across the flats

On an EVEN-tooth sprocket two tooth spaces sit exactly opposite each other, so a vernier across them reads the bottom diameter directly: PD − roller, and AmesWeb note that the even-tooth caliper diameter IS the bottom diameter. On an ODD-tooth sprocket no two spaces are opposite. The best you can do is measure across two that are nearly opposite, and the chord between them is PD·cos(90°/N), not PD — so the caliper diameter is PD·cos(90°/N) − roller. This batch checked that by construction rather than by trusting the formula: place N roller centres on the pitch circle, take the largest distance between any two of them, subtract a roller. For every even count from 6 to 64 that reproduces the first form and for every odd count the second, to twelve decimal places. The error column is what you read if you use the wrong one: 0.24 mm on 21 teeth of chain 40, and worse on fewer. That is the difference between accepting a sprocket and rejecting it.
Teeth (all odd)	Pitch Ø (mm)	What the even-tooth form would give (mm)	Correct odd-tooth caliper Ø (mm)	Error (mm)	Error (%)
9	37.132	29.208	28.643	0.564	1.969
11	45.078	37.153	36.695	0.459	1.250
13	53.068	45.143	44.756	0.387	0.865
15	61.084	53.159	52.824	0.335	0.633
17	69.116	61.191	60.896	0.295	0.484
19	77.159	69.234	68.971	0.264	0.382
21	85.211	77.286	77.048	0.238	0.309
25	101.330	93.405	93.205	0.200	0.215
31	125.533	117.609	117.447	0.161	0.137
41	165.906	157.981	157.860	0.122	0.077
57	230.541	222.616	222.529	0.088	0.039
95	384.111	376.186	376.134	0.053	0.014

Roller diameters, both families — the number that sets the root and caliper diameters

The pitch and outside diameters on this page do not use the roller at all; the root and caliper diameters use nothing else. So this is the column that decides whether a sprocket fits a chain. Read the two 12.70 mm rows together: ANSI 40 has a 7.92 mm roller and ISO 08B-1 an 8.51 mm one, which is 7.4% larger, so on a 21-tooth sprocket their root diameters are 77.29 mm and 76.70 mm. Same pitch, different tooth pocket, not interchangeable. The roller ÷ pitch column shows the two families’ different proportions: ANSI runs about 0.62 to 0.63 and ISO about 0.63 to 0.67, with the small ISO sizes proportionally fattest. The ANSI 25 and 35 rows are bushing diameters — the trailing 5 in the chain number means the chain has no rollers — and ANSI 41’s trailing 1 means lightweight. 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.
Chain	Pitch (mm)	Roller Ø (mm)	Roller ÷ pitch	Note
ANSI 25	6.35	3.30	0.5200	bushing, no rollers
ANSI 35	9.525	5.08	0.5333	bushing, no rollers
ANSI 40	12.7	7.92	0.6240	—
ANSI 41	12.7	7.77	0.6120	lightweight
ANSI 50	15.875	10.16	0.6400	—
ANSI 60	19.05	11.91	0.6253	—
ANSI 80	25.4	15.88	0.6250	—
ANSI 100	31.75	19.05	0.6000	—
ANSI 120	38.1	22.22	0.5833	—
ANSI 140	44.45	25.40	0.5714	—
ANSI 160	50.8	28.57	0.5625	—
ISO 04B-1	6	4.00	0.6667	—
ISO 05B-1	8	5.00	0.6250	—
ISO 06B-1	9.525	6.35	0.6667	—
ISO 08B-1	12.7	8.51	0.6701	—
ISO 10B-1	15.875	10.16	0.6400	—
ISO 12B-1	19.05	12.07	0.6336	—
ISO 16B-1	25.4	15.88	0.6252	—
ISO 20B-1	31.75	19.05	0.6000	—

What this page does NOT reproduce, and why

This table is the page’s boundary, written down. Everything above it is derived from three numbers — pitch, tooth count and roller diameter — by formulas that ASME B29.1 publishes and that two independent catalogues print in the same form. Everything in it needs either a copyrighted table or a particular maker’s product decision, and a page that guessed at those would be worse than a page that says so.
What	Where it lives	Why it is not here
The tooth form itself — flank radius, roller seat radius, topping radius	ASME B29.1, Figure 6 “Theoretical Tooth Form”, with the numeric data in its Tables 7 and 7M	Copyrighted, and it is a form rather than a dimension. Every figure this page prints can be derived from the pitch, the tooth count and the roller; the tooth form cannot, and reproducing it from a redrawn diagram would be guessing at a curve.
The pressure angle of the tooth	Implied by the same tooth form; it is not a single number but a function of the tooth count	It falls out of the flank geometry, so it is unavailable for the same reason. Where a catalogue prints a single pressure angle it is quoting one tooth count.
Caliper diameter TOLERANCES	ASME B29.1 Tables 12, 12M, 13 and 13M	The nominal caliper diameters here are computable and are computed. The tolerance bands around them are tabulated in the standard and were not obtainable. A measured caliper diameter has to be judged against those tables, not against the nominal.
The maximum BORE	The maker’s catalogue for the particular sprocket	The maximum HUB and groove diameter is a B29.1 formula and is printed above. The maximum bore is not: it depends on the hub wall the maker leaves, on the keyway, and on whether the sprocket is plate, hubbed or taper-bushed. This page will not invent it.
A maximum sprocket speed or a power rating	The chain maker’s rating tables	Neither is geometry. For chain selection see the roller chain selection calculator, which explains why it carries no rating curve either.

Six diameters from three numbers, and the one that gets measured wrong

A sprocket drawing needs six diameters and this page computes all six from three numbers. Pitch diameter is exact and nobody argues about it: N chords of length p inscribed in a circle give p / sin(180°/N). Outside diameter is ASME B29.1’s p(0.6 + cot 180°/N), which Martin Sprocket print verbatim as OD = (Pitch)(.6 + COT[180/Nt]) and ISO 606 writes identically as dₐ = p(0.6 + cot π/z) — the two families differ about many things and not about this. Root diameter is the pitch diameter minus the roller. And then there are the two caliper diameters, which is where sprockets get measured wrong.

The odd-tooth caliper diameter is the reason this page exists. On an even-tooth sprocket, two tooth spaces sit exactly opposite each other, so a vernier laid across the flats reads the root diameter directly — AmesWeb note that for an even count the caliper diameter IS the bottom diameter. On an odd-tooth sprocket no two spaces are opposite. The best you can do is span two that are NEARLY opposite, and the chord between those two roller seats is PD·cos(90°/N), not PD. So the caliper diameter is PD·cos(90°/N) − roller, and on 21 teeth of chain 40 that is 0.24 mm less than the even-tooth figure. Measure an odd sprocket against the wrong number and you will reject a good part or accept a bad one. This batch did not take the formula on trust: it placed N roller centres on the pitch circle, took the largest distance between any two of them, subtracted a roller, and recovered both forms — the even one and the odd one — for every tooth count from 6 to 64, to twelve decimal places.

Three things get worse together as the tooth count falls, and the chart shows two of them. The chordal speed variation is 1 ÷ cos(180°/N) − 1, because a chain leaves a sprocket as a polygon rather than a circle: 1.73% at 17 teeth, 3.53% at 12, 6.42% at nine. That is a cyclic load on every bearing and every coupling downstream, and it is why Renold Jeffrey want 17 teeth minimum and Diamond-Drives want 25 at high speed. The error in the PD + 0.6p shortcut is p·tan(90°/N), so it too is worst on a small sprocket — exactly where a drawing has least clearance to give away. And the maximum hub and groove diameter, p(cot 180°/N − 1) − 0.030 in, collapses: on nine teeth of chain 40 it is 21.4 mm, which after a bore and a keyway is not much sprocket at all.

What is not here, and why not. The tooth form itself — flank radius, roller seat radius, topping — is drawn in B29.1’s Figure 6 and tabulated in its Tables 7 and 7M, and it is copyrighted. It is also not a dimension but a form, so there is no honest way to reproduce it from a redrawn picture. The same goes for the pressure angle, which is a function of the tooth count rather than a number. The caliper diameter TOLERANCES in Tables 12, 12M, 13 and 13M are likewise not here, which matters: the nominal caliper figures above tell you what to expect, and only those tables tell you how far off is too far. The maximum BORE is refused for a different reason — it is not in the standard at all. It depends on the hub wall a particular maker leaves, on the keyway, and on whether the sprocket is plate, hubbed or taper-bushed, so it belongs in a catalogue and not in a formula. The maximum HUB and groove diameter IS a B29.1 formula and is printed.

And the roller column is what decides whether a sprocket fits a chain. Pitch and outside diameter do not use the roller; root and caliper use nothing else. ANSI 40 and ISO 08B-1 are both 12.70 mm pitch and their rollers are 7.92 mm and 8.51 mm, so on 21 teeth their root diameters are 77.29 mm and 76.70 mm — a 0.59 mm difference in the tooth pocket. That is the whole answer to whether a metric chain will run on an inch sprocket. For the drive as a whole, the chain length and centre distance calculator lays it out; for which chain in the first place, the roller chain selection calculator; for when a worn chain means the sprocket goes too, the chain wear and elongation calculator; and for the service factor that sizes all of it, the drive service factor calculator.

Frequently asked questions

Why are there two caliper diameters?

Because parity changes the geometry. On an even-tooth sprocket two tooth spaces are diametrically opposite, so the caliper spans a full diameter and reads PD − roller, which is also the root diameter. On an odd-tooth sprocket nothing is opposite anything; the two seats nearest to opposite are separated by PD·cos(90°/N), so the caliper reads PD·cos(90°/N) − roller. On 21 teeth of chain 40 those are 77.29 mm and 77.05 mm. Use the even figure on an odd sprocket and you are 0.24 mm out, which on an inspection sheet is the difference between pass and fail.

Should I use an odd or an even number of teeth?

Odd, for wear, if nothing else decides it. With an odd tooth count and an even link count, a given chain roller meets a different tooth on successive revolutions, so wear spreads round the sprocket instead of concentrating. The cost is that you cannot put a vernier across the flats and read the root diameter — you have to use the odd-tooth caliper figure. That is an inspection inconvenience, not an engineering objection.

Why is the outside diameter not just PD + 0.6p?

Because the two expressions differ by exactly p·tan(90°/N). ASME B29.1’s form is p(0.6 + cot 180°/N); subtract PD = p/sin(180°/N) from it and the remainder is p(0.6 − tan(90°/N)), not 0.6p. So PD + 0.6p is always too big, by 0.95 mm on 21 teeth of chain 40, 2.24 mm on nine teeth and 0.17 mm on 120. It is an approximation that is good where you do not need it and poor where you do. Worth adding that B29.1’s own form is itself called the APPROXIMATE outside diameter: the true tip is set by the tooth form’s topping radius, which is in the standard’s Figure 6 and is not reproduced here.

What is chordal action and why does it set the minimum tooth count?

A chain wraps a sprocket as a polygon, not a circle. The roller that is currently leaving the sprocket moves at the sprocket’s rim speed, but only the component along the straight run reaches the chain — and that component varies as the roller swings through one pitch angle. The result is that the chain’s speed varies by 1 ÷ cos(180°/N) every tooth: 1.73% at 17 teeth, 3.53% at 12, 6.42% at nine. This batch confirmed it by simulation rather than by algebra — rotate the sprocket, search for the roller at the departure point, project its velocity on the run, and check that the mean speed still pays out N pitches per revolution. That ripple is a cyclic load on everything downstream, which is why 17 teeth is the usual floor and 25 the figure for a fast drive.

Can I put a metric chain on an inch sprocket of the same pitch?

No. ANSI 40 and ISO 08B-1 are both 12.70 mm pitch and that is where the agreement ends. Their rollers are 7.92 mm and 8.51 mm, so the tooth pockets are cut to different radii and the root diameters differ by 0.59 mm on a 21-tooth sprocket. Their inner widths differ too — 7.92 mm against 7.75 mm. The chain will go on and it will run, and it will wear both parts far faster than either was designed for, because the rollers will not seat. Five catalogues agree on every one of those dimensions, so this is not a case where the sources are in doubt.

What is the maximum bore for this sprocket?

This page will not tell you, and the reason is that the standard does not either. What ASME B29.1 gives is the maximum HUB and groove diameter, p(cot 180°/N − 1) − 0.030 in, and that is computed above — 70.8 mm at the defaults. The maximum BORE is a product decision: it depends how much hub wall the maker leaves, on the keyway, and on whether the sprocket is a plate, a hubbed casting or a taper bush. Two makers’ 21-tooth chain-40 sprockets can honestly carry different maximum bores. Take that figure from the catalogue page for the sprocket you are buying.

Does the page work for ISO 606 sprockets as well as ANSI?

Yes, and with the same formulas. MechanixCalc print ISO 606’s sprocket relations as d = p/sin(π/z), dₐ = p(0.6 + cot π/z) and d_f = d − d_roller, which are the ANSI forms with different letters. What you must change is the roller diameter, which the chain-size selector does for you: pick an ISO designation and the roller field fills with the ISO figure. The one thing the ISO family does not give you is the maximum hub and groove formula, which is a B29.1 expression in inches — it is printed here for both, and for an ISO chain it should be treated as indicative.

Related calculators

References

ASME B29.1-2011 (R2022), Precision Power Transmission Roller Chains, Attachments, and Sprockets. Cited by clause and figure; the standard itself is copyrighted and was not fetched. Its published preview at webstore.ansi.org confirms the structure this page relies on: section 1 Roller Chain, section 3 Sprockets, Figure 6 Theoretical Tooth Form, Figure 7 Sprocket Diameters, Tables 12/12M and 13/13M (caliper diameter tolerances) and Table 14 (pitch and outside diameter). Every printed dimension used here is attributed to a named catalogue instead, and everything computable is computed.
AmesWeb. Roller Chain Sprocket Diameter Calculator — ASME B29.1. The one source consulted that prints all five diameter relations in one place: pitch diameter P / sin(180/N); bottom diameter PD − Dr; caliper diameter (even teeth) = bottom diameter; caliper diameter (odd teeth) = PD × cos(90/N) − Dr; “approximate outside diameter” = P × (0.6 + cot(180/N)); and maximum hub and groove diameter = P × (cot(180/N) − 1) − 0.030 in. AmesWeb also notes that “Dr is bushing diameter for chain 25 and chain 35 as these chains have no rollers”.
Martin Sprocket & Gear. Sprocket Engineering Data, catalogue section E-152. Prints the B29.1 forms directly: “PD = Pitch / SIN (180/Nt)” and “OD = (Pitch)(.6 + COT [180 / Nt])”, and describes the bottom diameter as “the diameter of a circle tangent to the bottoms of the tooth spaces”. Also the source, with chinatransmissions’ ANSI chart, for the roller diameters and roller widths used here.
MechanixCalc. ISO 606 Calculator — Roller Chain Selection & Sprocket Geometry. Prints the ISO 606 sprocket forms d = p/sin(π/z), da = p(0.6 + cot(π/z)) and df = d − d_roller, so the tip-diameter form is the same in the ISO family as in the ANSI one; and states the selection route used here — “the chain is selected by comparing the design load (nominal power × service factor) against the chain’s minimum breaking load divided by a speed-dependent safety factor”.
chinatransmissions. ANSI Roller Chain Size Chart | Dimension Guide. Second, independent transcription of the ANSI roller diameters and roller widths, agreeing with Martin’s where the two overlap. It is also the source that settles chain 41’s roller diameter at 0.306 in — the page being ported had 6.35 mm, which is 0.250 in, chain 41’s roller WIDTH.
ISO 606:1994, Short-pitch transmission precision roller and bush chains, attachments and associated chain sprockets (second edition, 1994-02-15). Cited by number; only the standard’s preview was reachable, which confirms clause 3.6 (a measuring force applied when a chain’s length is measured) and clause 5 (chain wheels, described in its own foreword as ‘the unification of all the relevant national Standards’). The dimensional figures here are taken from five named catalogues instead — see the breaking-load note.
iwis / JWIS. Catalogue: precision chains, British Standard roller chain table. The lowest of the five minimum-breaking-load columns compared in this batch, and the one that matches the figures usually quoted as ISO 606’s own minima (16B-1 at 60.0 kN, 20B-1 at 95.0 kN). Used as the default on the selection page.
R.S. Khurmi and J.K. Gupta, A Textbook of Machine Design, chapter 21 Chain Drives and chapter 20 V-Belt and Rope Drives (the chapters as distributed by Al-Mustansiriyah University and Al-Mustaqbal University). Source for Table 21.2, factor of safety for bush roller chain against the smaller sprocket’s speed; for the service factor as K1 × K2 × K3 (load, lubrication, hours); for the recommended teeth on the smaller sprocket against velocity ratio; for the V-belt tension ratio 2.3 log(T1/T2) = μθ cosec β with β the groove HALF angle and a groove angle of 32° to 38°; for the centrifugal tension Tc = mv²; and for the maximum-power condition Tc = T/3.
Diamond-Drives by Timken. Frequently Asked Questions. Maximum allowable wear elongation “approximately 3% for most industrial applications, based upon sprocket design” and “approximately 1.5%” where centres are fixed or the drive must run smoothly; the 200/N relationship, N being the teeth in the large sprocket; and minimum tooth counts against speed — 12 slow, 17 medium, 25 high. Its stated range of application for 200/N contradicts Reliable Plant’s; see the note on the wear page.