Spur Gear Geometry Calculator

Every dimension a spur gear drawing needs — pitch, tip, root and base diameters, centre distance, tooth thickness and depths — plus the undercut limit computed rather than quoted, and the span and over-pins measurements that let you check a gear you did not cut.

Last updated 24 Sep 2026

Spur gear geometry

Module, teeth and pressure angle → every diameter

Module m (mm)Tooth size. The pitch diameter is m×z, the whole tooth depth is 2.25 m and the centre distance is m(z₁+z₂)/2 — so every dimension on the page scales with it. ISO 54 series 1 is 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10…

Teeth on the pinion z₁The small gear. This is the one that undercuts, the one that fails in bending and the one that wears out, because every tooth on it meets the load z₂/z₁ times as often.

Teeth on the wheel z₂The large gear. For the geometry of the pinion alone this only affects the centre distance and the ratio.

Pressure angle α20° is the modern standard and what almost every stock gear is cut to. 14.5° is the legacy American system and a 14.5° gear WILL NOT MESH with a 20° one of the same module — they are different tooth forms, not different tolerances.

Teeth to span, for the base tangent measurementHow many teeth the anvils of a gear-tooth micrometer straddle. Pick the recommended value below: too few and the anvils land near the tip where the flank has run out, too many and they touch the tips instead of the flanks.

Pin or ball diameter for the over-pins measurement (mm)Two pins dropped into opposite tooth spaces and measured across. The “best size” is 1.68 m at 20° and 1.92 m at 14.5°, which puts the contact near the pitch circle so the answer is least sensitive to the pressure angle being slightly off.

Not a circuit: a gear tooth drawn at TWO scales at once, because it has to be. The main figure is one tooth with its radial scale magnified about nine times — one module is 34 units of the drawing, so the 2.25 m whole depth is visible — and the little gear at the bottom left is the same tooth on the same gear at true scale, which is why the magnification is needed at all. All five diameters are marked as horizontal levels. Four of them are fixed multiples of the module and do not move; the BASE circle does, because its depth below the pitch circle is z(1 − cos α)/2 modules, so it climbs with the tooth count and with the pressure angle. Watch where it goes: on a 20-tooth 20° gear the base circle sits 0.60 m below the pitch line, well above the root — but above about 42 teeth it falls below the root circle entirely, and the whole flank is then a true involute. When the pinion is undercut the tooth is redrawn with its root scooped away, which is what the generating tool actually does to the section that carries the bending moment.

60.000mmExample

A 2 mm module 20/40 pair at a 20° pressure angle, with the span taken over three teeth and 3.36 mm pins

Three choices, everything else follows

d = m·z · dₐ = m(z + 2) · d_f = m(z − 2.5) · d_b = d·cos α · a = m(z₁ + z₂)/2 · s = πm/2 · z_min = 2/sin²α · W_k = m·cos α·[π(k − 0.5) + z·inv α]

m: module, in millimetres. The whole page scales with it linearly
z: number of teeth, which must be an integer — this page rounds it
α: pressure angle. 20° by default; 14.5° and 25° are different tooth forms and do not mesh with it
2, 2.5: twice ISO 53’s addendum coefficient (1 m) and twice its dedendum coefficient (1.25 m). The 0.25 m difference is bottom clearance, which is what stops the mating tip bottoming in the root
inv α: the involute function, tan α − α with α in radians. 0.0149044 at 20° and 0.0055448 at 14.5°
z_min: the tooth count below which the generating tool cuts away part of the working flank. NOT an integer: 17.0973 at 20°

Worked example

A 2 mm module 20/40 pair at a 20° pressure angle, with the span taken over three teeth and 3.36 mm pins
The centre distance first, because it is the dimension the housing has to hold: a = m(z₁ + z₂)/2 = 2(20 + 40)/2 = 60 mm. Note that this is fixed by the module and the tooth counts alone — for a spur pair there is no freedom in it at all, which is why a housing dimension that has to be something else forces either a profile shift or a helical pair
The pitch diameters: d₁ = 2×20 = 40 mm and d₂ = 2×40 = 80 mm. The tip diameter, which is what you turn the blank to, is m(z+2) = 44 mm, and the root diameter is m(z−2.5) = 35 mm. The whole depth is 2.25 m = 4.50 mm
The base diameter is the one that is not obvious: d_b = d·cos 20° = 37.5877 mm. It sits between the root and the tip, and the flank is only a true involute outside it. That is the whole of the undercut question
The undercut limit. 2/sin²20° = 17.0973 teeth, so the integer minimum is 18. Your 20-tooth pinion clears it by 2.903 teeth. Had it been 17, the profile shift needed would be 1 − z/z_min = 0.0057, which is 0.0114 mm of cutter withdrawal — undetectable, which is why 17 is quoted as the limit everywhere despite being below it
Tooth thickness on the pitch circle is πm/2 = 3.1416 mm, exactly half the circular pitch πm = 6.2832 mm. The base pitch, measured along the line of action, is πm·cos α = 5.9043 mm, and it is the base pitch and not the circular pitch that decides the contact ratio
Now the two measurements that let you check a real gear. The base tangent measurement over three teeth is m·cos α[π(3 − 0.5) + 20·inv 20°] = 15.3209 mm. The same gear cut at 14.5° would span 15.4224 mm, which is -0.7 per cent different — easily enough to tell the two tooth forms apart with a gear-tooth micrometer
And the measurement over pins. Dropping 3.36 mm pins (1.68 m, the best size at 20°) into opposite spaces, the contact angle φ satisfies inv φ = inv α + d_p/(m·z·cos α) − π/2z, which gives φ = 23.827° and M = m·z·cos α/cos φ + d_p = 44.4498 mm. That is 0.450 mm outside the tip diameter, so the micrometer clears the blank, which is the point of the method. On an ODD tooth count the pins are not diametrically opposite and the reading has to be multiplied by cos(π/2z) — a 21-tooth gear here reads 46.3331 mm rather than the 46.4536 mm the even-count formula would give

Every dimension a spur gear drawing needs, and where it comes from

Only three of these are choices — the module, the tooth count and the pressure angle. Everything else follows, and the four coefficients that make it follow (addendum 1 m, clearance 0.25 m, so dedendum 1.25 m and whole depth 2.25 m, with a 0.38 m root fillet) are ISO 53’s standard basic rack tooth profile. ISO 53:1998 specifies exactly ONE pair of values for the clearance and the fillet radius, which is why they are constants here and not a choice. The base diameter is the odd one out and the important one: it is the circle the involute is unwrapped from, it is where the line of action is tangent, and it is the reason two gears of the same module but different pressure angles cannot mesh. 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.
Dimension	From	For the example: m = 2, z = 20, α = 20°
Pitch diameter d	m·z	40.0000 mm
Tip (outside) diameter dₐ	m(z + 2)	44.0000 mm
Root diameter d_f	m(z − 2.5)	35.0000 mm
Base diameter d_b	d·cos α	37.5877 mm
Addendum hₐ	1 m	2.0000 mm
Dedendum h_f	1.25 m	2.5000 mm
Whole depth h	2.25 m	4.5000 mm
Root fillet ρ_f	0.38 m	0.7600 mm
Circular pitch p	πm	6.2832 mm
Base pitch p_b	πm·cos α	5.9043 mm
Tooth thickness on the pitch circle s	πm/2	3.1416 mm
Centre distance a	m(z₁ + z₂)/2	60.0000 mm

The undercut limit, and the 17 that is really 17.0973

Read the second and third columns together, because the difference between them is the single most-repeated rounding error in gearing. 2/sin²α at 20° is 17.0973, and the integer minimum is therefore 18 — which is exactly what SDP/SI’s own catalogue prints: “for 14.5° the value of z_c is 32, and for 20° it is 18”. The 17 that circulates everywhere is 17.0973 truncated, which is the wrong direction for a minimum. It survives because at 17 teeth the undercut is tiny: the profile shift needed to clear it is 0.0057 m, which on a 2 mm module is eleven microns of cutter withdrawal, and tec-science reports that in practice “a minimum number of teeth of 14 is assumed”. So the honest answer is that 18 is the limit, 17 is fine, and 14 works if you can live with a visibly scooped root. The last two columns carry the identity that makes the whole thing hold: a real hob cuts 1.25 m deep, not 1 m, and a sharp corner at that depth would need 21.4 teeth — but the 0.38 m tip fillet lifts the generating corner back up by 0.38(1−sin α), which at 20° is 0.2500 m, cancelling the clearance exactly. At 20° the effective depth is one module to four figures. At 14.5° it is 0.9651, so the published 32 is about three per cent conservative there.
Pressure angle	2/sin²α	Integer minimum	If the tool corner were sharp at 1.25 m	Effective hob depth (×m)
14.5°	31.903	32	39.88	0.9651
17.5°	22.118	23	27.65	0.9843
20.0°	17.097	18	21.37	1.0000
22.5°	13.657	14	17.07	1.0154
25.0°	11.198	12	14.00	1.0306
30.0°	8.000	8	10.00	1.0600

Three pressure angles, and why they do not mesh with each other

This page assumes whichever pressure angle you selected, and the selection matters more than it looks. A 20° gear and a 14.5° gear of the same module have the same pitch diameter, the same tip diameter, the same root diameter and the same tooth count — and they do not mesh. The flanks are involutes of different base circles, so they touch at a point rather than rolling, and a pair like that runs rough, loads one corner of each tooth and wears out quickly. It is the commonest expensive mistake in buying a replacement gear, because nothing about the outside of the part tells you which form it is. Measure it: the base tangent measurement below differs by several per cent between the two forms on the same nominal gear, which is how you tell. 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.
Tooth form	Where you meet it	Minimum teeth free of undercut	What you trade
20° full depth	The modern default. ISO 53’s standard basic rack, DIN 867, and what essentially every stock gear catalogue sells	17.1 — so 18, or 17 with undercut nobody can measure	Lower contact ratio than 14.5°, stronger tooth, less sensitive to centre-distance error, and a bigger radial separating force
14.5° full depth	The legacy American system. Still met on old machine tools, clock and instrument gearing, and replacement parts	31.9 — so 32, which is why small 14.5° pinions barely exist	Higher contact ratio and therefore quieter, but a weaker tooth and a much worse undercut limit. Its radial force is smaller, which is why it survived in light instrument drives
25° full depth	Used where tooth strength or surface durability governs — heavy industrial and some automotive gearing	11.2 — so 12	Strongest tooth of the three and the best undercut limit, at the cost of a lower contact ratio, more noise and the largest separating force of the three

Base tangent measurement over k teeth, and how it distinguishes the two tooth forms

For a m = 2, z = 20 gear. The span is derived rather than quoted: it is (k−1) base pitches plus one base-circle tooth thickness, which collects into W_k = m·cos α·[π(k − 0.5) + z·inv α]. Two things make it the measurement to use. It is independent of the centre distance and of the tip diameter, so it measures the TOOTH rather than the blank; and because it carries cos α and inv α it is a direct test of the pressure angle. A gear you believe is 20° that spans like the third column is 14.5°. The recommended k puts the anvils near mid-tooth-height; span too few teeth and they land where the flank has run out into the tip chamfer, span too many and they touch the tips instead of the flanks and the reading means nothing.
Teeth spanned k	W_k at 20° (mm)	W_k at 14.5° (mm)	Difference
2	9.4166	9.3393	0.8%
3	15.3209	15.4224	-0.7%
4	21.2251	21.5054	-1.3%
5	27.1294	27.5885	-1.7%
6	33.0337	33.6715	-1.9%

Three choices, every diameter, and the 17 that is really 17.0973

Three numbers decide a spur gear and everything else follows. The module sets the tooth size, the tooth count sets the diameter, and the pressure angle sets the tooth form. Pitch diameter is m·z; tip diameter is m(z + 2) because the addendum is one module; root diameter is m(z − 2.5) because the dedendum is 1.25 modules, the extra quarter being bottom clearance so the mating tip does not bottom in the root. Centre distance is m(z₁ + z₂)/2 and — this is worth noticing — for a spur pair there is no freedom in it at all. If the housing has to be something else, you need either a profile shift or a helical pair, where the helix angle buys you the centre distance. The four coefficients come from ISO 53’s standard basic rack tooth profile, which specifies exactly one pair of values for the clearance and the root fillet.

The base diameter is the one that matters and the one nobody draws. d_b = d·cos α. It is the circle the involute is unwrapped from, the line of action is tangent to it, and the flank is only a true involute outside it. Two consequences. Two gears of the same module and different pressure angles have identical pitch, tip and root diameters and CANNOT MESH, because their flanks are involutes of different circles — a 14.5° gear run against a 20° one touches at a point instead of rolling on a line. And the base pitch, πm·cos α, is what sets the contact ratio, not the circular pitch.

The undercut limit is 17.0973, not 17. The generating tool cuts away part of the working flank when the tooth count falls below 2/sin²α — 17.0973 at 20°, 31.9029 at 14.5°, 11.1978 at 25°. So the integer minimum at 20° is 18, which is what SDP/SI’s catalogue prints, and the 17 everybody quotes is that number truncated in the wrong direction for a minimum. It survives because it barely matters: the profile shift needed to clear the limit at 17 teeth is 0.0057 modules, eleven microns on a 2 mm module, and tec-science reports 14 teeth as the practical floor. But the direction of the rounding is worth knowing, and so is the reason the criterion uses a one-module tool depth when a real hob cuts 1.25 m: the hob’s 0.38 m tip fillet lifts the generating corner back up by 0.38(1 − sin α), which at 20° is 0.2500 m — exactly the clearance. ISO 53’s two constants cancel. At 14.5° they do not quite, and the published 32 is about three per cent conservative.

Two measurements you can actually take. A gear you did not cut has to be measured, and neither the pitch diameter nor the tooth thickness is directly measurable — the pitch circle is imaginary and the tooth thickness is an arc. The base tangent (span) measurement over k teeth solves it: W_k = m·cos α[π(k − 0.5) + z·inv α], taken with a flat-anvil gear-tooth micrometer, independent of the centre distance and of the tip diameter. The measurement over two pins does the same job with a plain micrometer and two gauge pins, at the cost of needing the involute function inverted. Both are on the page, and the span is the practical way to tell a 20° gear from a 14.5° one: the same nominal gear spans several per cent differently in the two forms.

What this page does not do. It computes standard, unshifted teeth. It will tell you the profile shift you need to clear undercut, but it does not carry a shifted pair’s operating pressure angle, its altered centre distance or its tip shortening — those need the whole shifted-pair calculation and a decision about how the shift is split between pinion and wheel. It is also a geometry page and not a strength page: for a first bending number see the gear tooth bending stress calculator, for the mesh continuity see the contact ratio and backlash calculator, and for the shaft the gear sits on, the parallel key and keyway calculator and the shaft deflection and slope calculator.

Frequently asked questions

What is the formula for the outside diameter of a spur gear?

dₐ = m(z + 2) in metric, because the addendum of the standard ISO 53 basic rack is one module, so the tip circle sits one module outside the pitch circle on each side. For a 2 mm module 20-tooth gear that is 2(20 + 2) = 44 mm. In diametral pitch terms it is (z + 2)/P_d. Note that this is the number you turn the blank to, and that it is the one gear dimension with a generous tolerance — the tip circle carries no load and does not locate anything.

Why is the minimum number of teeth 17?

It is not quite. The limit is 2/sin²α, which at 20° is 17.0973, so the smallest integer free of undercut is 18 — and that is what SDP/SI’s own catalogue prints. The 17 that circulates everywhere is 17.0973 truncated, which is the wrong direction for a minimum. In practice 17 is used constantly, because the amount of undercut at 17 teeth corresponds to 0.0057 modules of profile shift, which on a 2 mm module is about ten microns and nobody can see it. Below roughly 14 teeth the undercut becomes real, and pinions that small are designed as profile-shifted gears from the start.

Will a 14.5° gear mesh with a 20° gear?

No, and this is the expensive one. Two gears of the same module and tooth count have the same pitch diameter, the same tip diameter and the same root diameter whatever the pressure angle, so nothing about the outside of the part gives it away. But the base diameter is d·cos α, so the flanks are involutes of different circles: the pair touches at a point instead of rolling on a line, runs rough, concentrates the load at one corner of each tooth and wears out. Measure the base tangent span before ordering a mate — it differs by several per cent between the two forms on the same nominal gear.

How do I measure a gear’s tooth thickness without a gear-tooth vernier?

Two pins and a plain micrometer. Drop equal gauge pins into two opposite tooth spaces and measure across them: the reading is m·z·cos α/cos φ + d_p, where φ comes from inverting the involute function on inv α + d_p/(m·z·cos α) − π/2z. On an odd tooth count multiply by cos(π/2z), because the two pins are then not diametrically opposite. This page does both. Use the “best size” pin — 1.68 m at 20° — which puts the contact near the pitch circle where the reading is least sensitive to the pressure angle.

What is the centre distance for a spur gear pair?

a = m(z₁ + z₂)/2, and for standard unshifted teeth there is no freedom in it whatever. That is the single biggest practical difference between spur and helical gearing: a helical pair can hit almost any centre distance by choosing the helix angle, because its centre distance is m_n(z₁ + z₂)/(2·cos β). If a spur pair has to fit a centre distance that the tooth counts do not give, the options are a profile shift on both gears, a different module, or different tooth counts.

Why is the dedendum 1.25 m and not 1 m?

The extra 0.25 m is bottom clearance, and ISO 53 specifies exactly that value. Without it the mating gear’s tip would bottom in the root as soon as the centre distance was a little tight, which jams the pair and is far worse than the small loss of tooth height. The clearance is also where the root fillet lives — 0.38 m for the standard basic rack — and that fillet is what carries the bending stress. It has a third consequence that almost nobody mentions: 0.38(1 − sin 20°) is 0.2500, so a standard hob’s tip fillet lifts its generating corner back up by exactly the clearance, which is why the undercut limit comes out as 2/sin²α and not 2.5/sin²α.

Can I use this page for a profile-shifted gear?

Only for the shift it recommends. The page computes the profile shift coefficient you need to clear undercut, which is 1 − z/z_min, and it prints that in millimetres of cutter withdrawal. It does NOT carry the rest of a shifted pair’s geometry: the operating pressure angle, the increased centre distance, the tip shortening needed to keep the clearance, or the way the total shift should be split between pinion and wheel. Those need the full shifted-pair calculation and a design decision this page cannot make for you.

Related calculators

References

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.
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.
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.
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.
tec-science, Undercut of gears. A second reading of the undercut limit z_min = 2/sin²α, and the source for the practical position that “in practice, however, a minimum number of teeth of 14 is assumed” — because a small amount of undercut below the active flank costs very little. Note that this page rounds the 20° result to 17 in its text, which is the truncation this site declines to repeat.
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.
ISO 1328-1:2013, Cylindrical gears — ISO system of flank tolerance classification. Cited by number and NOT reproduced. It is named here because a recommended backlash is meaningless without a flank tolerance class behind it, and because the accuracy grade is what decides how much of your backlash budget the gears themselves consume before the centre distance is considered.