Gear Profile Shift Calculator

The operating pressure angle, the centre distance modification coefficient and the tip shortening coefficient computed side by side — because they are three different numbers, and the centre distance does not grow by the sum of the shifts.

Last updated 24 Sep 2026

Gear profile shift

Shifts and tooth counts → centre distance, tips, top land

Module m (mm)Profile shift is measured in modules, so every coefficient on this page is dimensionless and the module only turns them into millimetres at the end. A shift coefficient of 0.5 means the cutting tool was withdrawn by half a module.

Pinion teeth z₁The small gear, and the one profile shift usually exists to rescue. Below 17.0973 teeth at 20° a rack-generated tooth is undercut; the spur gear geometry page computes that limit and the minimum shift that clears it, and this page deliberately does not repeat the calculation. Start there, then bring the number here.

Wheel teeth z₂The large gear. The sum z₁ + z₂ is what the operating pressure angle depends on, not the ratio — a given total shift on a 15/30 pair moves the pressure angle far more than the same shift on a 60/120 pair, because it is spread over four times as many teeth.

Profile shift coefficient x₁ on the pinionPositive means the tool was withdrawn by x·m, so the tooth is thicker at the root and thinner at the tip. This is the usual place to put shift: it clears undercut, it strengthens the tooth that is weaker and cycles more often, and it moves the contact away from the root where the specific sliding is worst.

Profile shift coefficient x₂ on the wheelSet it to −x₁ and the centre distance does not move at all: the pair is balanced, the operating pressure angle stays at the cutting angle, there is no tip shortening, and the shift has been used purely to move strength from one gear to the other. That is called an S0 pair and it is the single most useful thing on this page.

Cutting pressure angle αThe pressure angle the CUTTER has, which is a property of the tool and of the standard basic rack. It is not the angle the pair runs at once it has been shifted — that is the operating pressure angle, computed above, and the difference between the two is where most of this page comes from.

Minimum tip thickness you will accept (× module)The only figure this batch could verify in print is 0.2 m_n, from Gear Technology’s Nonstandard Tooth Proportions: below it the tip is a knife edge and, in a case-hardened gear, a brittle one, because the case wraps round from both flanks and meets. The widely quoted 0.25 could not be traced to a named publication, and practice for carburised gears is stricter still. It is an input here for that reason.

Not a circuit: a geometry. The drawing at the top left is two teeth of the SAME 15-tooth pinion, one cut with no profile shift and one with x = +0.5, superimposed on the same reference circle. They are the same curve. A profile shift does not change the involute — the base circle has not moved — it changes which part of that involute becomes the tooth, so the shifted tooth is the unshifted one slid outward along itself: fatter where it matters at the root, thinner at the tip, and running out of tip eventually. That one picture is the whole argument for and against positive shift. The bar across the middle is live and is the point of the page: it is the total shift in millimetres, split into the solid part that actually moves the centre distance (y·m) and the hatched remainder that has to come off the tips (Δy·m). They are never equal, and the hatched part is what almost every explanation of profile shift leaves out. The scale at the bottom is your pinion's tip thickness in modules, with the 0.2·m published floor marked.

0.4711× mExample

A module 2 pair, 15 and 30 teeth at 20°, with x₁ = 0.5 on the pinion and x₂ = 0.2 on the wheel

Three coefficients, and they are never equal

inv α_w = inv α + 2(x₁+x₂)tan α / (z₁+z₂) · a_w = a·cos α / cos α_w · y = (a_w − a)/m · Δy = (x₁+x₂) − y

x: the profile shift coefficient. The cutting tool is moved radially by x·m: outward for positive, which thickens the root and thins the tip. It is a manufacturing instruction, not a shape — the tooth is still a pure involute of the same base circle, and only which PART of that involute is used has changed
inv α: the involute function, tan α − α. It has no inverse in closed form, so α_w has to be solved for numerically. This page unrolls six Newton steps from a cube-root start, which is enough for twelve correct figures over every input it accepts — the proof is in the batch’s own assertions and walks the whole range
y: the centre distance modification coefficient. The centre distance grows by y·m and by nothing else. It is ALWAYS smaller than x₁+x₂ when the sum is positive, and that inequality is the most commonly missed fact about profile shift
Δy: the tip shortening coefficient, sometimes written k and called Kopfkürzung in the German literature. It is the amount, in modules, that each tip RADIUS must be reduced to keep the bottom clearance at the basic rack’s 0.25·m. It falls straight out of the clearance equation: c = m(0.25 − Δy)
α_w: the operating or working pressure angle: the angle the pair actually runs at, which is not the angle the cutter had. It is what the operating pitch circles roll on, and every consequence of a profile shift — the centre distance, the contact ratio, the separating force, the tooth thicknesses where they touch — follows from it
s_a: tooth thickness at the tip circle, from s(r) = 2r[s₀/d + inv α − inv α_r]. Proved here against a simulation of the rack generation itself: the tooth is whatever no position of the cutting rack removes, and the two agree to about a micron

Worked example

A module 2 pair, 15 and 30 teeth at 20°, with x₁ = 0.5 on the pinion and x₂ = 0.2 on the wheel
THE OPERATING PRESSURE ANGLE FIRST, because everything follows from it. inv α_w = inv 20° + 2(0.5+0.2)tan 20° / (15+30) = 0.0149044 + 0.0113240 = 0.0262279. Solving tan α_w − α_w = 0.0262279 by Newton gives α_w = 23.9647°. Note how far it has moved: nearly four degrees, from one seventh of a module of total shift
THE CENTRE DISTANCE. a = m(z₁+z₂)/2 = 2 × 45/2 = 45 mm. a_w = a·cos α / cos α_w = 45 × 0.93969 / 0.91380 = 46.2753 mm
AND NOW THE PART EVERYBODY GETS WRONG. The centre distance grew by 1.2753 mm. The sum of the shifts is 0.7, which in millimetres is 0.7 × 2 = 1.4 mm. Those are different numbers and they are supposed to be. y = 1.2753/2 = 0.63765, not 0.7
THE DIFFERENCE IS THE TIP SHORTENING. Δy = (x₁+x₂) − y = 0.7 − 0.63765 = 0.06235, which is 0.1247 mm off each tip RADIUS. Check what happens without it: the unshortened pinion tip would be d_a1 = m(z₁+2+2x₁) = 2(15+2+1) = 36.000 mm, and the clearance to the wheel's root would be 46.2753 − 36.000/2 − 55.800/2 = 0.3753 mm, against the 0.500 mm the basic rack asks for. Shorten by Δy·m and it comes back to exactly 0.500 mm
So the tip diameters are d_a1 = m(z₁ + 2 + 2x₁ − 2Δy) = 35.7506 mm and d_a2 = 64.5506 mm. Those are the numbers that go on the drawing, and they are the ones a stock-gear catalogue cannot give you because they depend on the MATE
THE TIP THICKNESS, which is what stops you. The pinion's tooth thickness at its reference circle is s₁ = m(π/2 + 2x₁ tan α) = 3.8695 mm, up from 3.1416 for an unshifted tooth. Carried out to the tip circle through s(r) = 2r[s₁/d₁ + inv α − inv α_a1], with α_a1 = arccos(d_b1/d_a1) = 37.951°, it gives s_a1 = 0.9421 mm = 0.4711 modules. Comfortably above the 0.2 m_n that Gear Technology gives as the minimum, and above the 0.25 most people work to
WHAT THE SHIFT BOUGHT. A 15-tooth pinion at 20° is below the 17.0973 undercut limit, so an unshifted one is undercut and x₁ = 0.5 clears it with a large margin. The tooth is 23.2 per cent thicker at the reference circle. And it cost 0.1247 mm off both tips, 0.1247 mm of centre distance you did not get, and a pair that will now only mesh correctly at 46.275 mm and nowhere else

The three coefficients, for a module 2, 15/30 pair at 20°

Columns six and seven are the page. They are never the same number, and the gap between them is column four multiplied by the module. The reason is that the involute function is not linear: opening the centre distance raises the operating pressure angle, and the tooth thicknesses that the shift added grow the required opening less than proportionally. At a total shift of 1.0 on this pair, y is 0.8838 — the centre distance grows by 1.768 mm and not by the 2.000 mm the sum of shifts suggests, and the missing 0.232 mm has to come off the tips or the tips will hit the mating roots. Watch the last column fall below 0.5 mm, which is the 0.25·m the basic rack asks for: that is the clearance the shift ate, and it is why tip shortening exists. Note also the first row, and what happens further down: at a large enough NEGATIVE total shift there is no solution at all, because inv α_w would have to be negative and the involute function is not. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
x₁ + x₂	Operating α_w (°)	y	Δy	a_w (mm)	Centre distance grew by y·m	…what x·m would have said	Clearance if the tips were NOT shortened (mm)
-0.40	16.6293	-0.43404	0.03404	44.1319	-0.8681	-0.8000	0.4319
-0.20	18.4793	-0.20747	0.00747	44.5851	-0.4149	-0.4000	0.4851
0.00	20.0000	0.00000	0.00000	45.0000	0.0000	0.0000	0.5000
0.20	21.3042	0.19388	0.00612	45.3878	0.3878	0.4000	0.4878
0.40	22.4533	0.37739	0.02261	45.7548	0.7548	0.8000	0.4548
0.70	23.9647	0.63765	0.06235	46.2753	1.2753	1.4000	0.3753
1.00	25.2875	0.88382	0.11618	46.7676	1.7676	2.0000	0.2676
1.40	26.8366	1.19511	0.20489	47.3902	2.3902	2.8000	0.0902
2.00	28.8298	1.63440	0.36560	48.2688	3.2688	4.0000	-0.2312

What profile shift is for, in the order it is used

The order is ANSI/AGMA 901-A92’s, as Gear Technology reports it: avoid undercut, balance specific sliding, balance flash temperature, balance bending fatigue life, avoid narrow top lands. Two honest limits on this page. First, SPECIFIC SLIDING is not computed here. It needs the path of contact — the sliding at any contact point depends on how far that point is from the pitch point along the line of action — which belongs to the contact ratio page, and the published forms of the specific sliding ratio differ between sources in which velocity they normalise by. Computing it from a form this batch could not settle would be worse than not computing it. Second, FLASH TEMPERATURE is a scuffing criterion that needs a load, a surface speed, a lubricant and a friction coefficient, and it is an ISO/TS 6336-20 calculation rather than a geometric one. Both are named here so that a reader knows the page is not claiming to have balanced them. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Purpose	How often	Which way the shift goes	What it costs
1. Clearing undercut on a small pinion	Very common. It is what profile shift was invented for and it is still the commonest reason to use it	POSITIVE, on the pinion. The tool is withdrawn so its tip no longer cuts into the flank below the base circle	Thins the pinion’s tip and raises the operating pressure angle, which raises the separating force on the bearings
2. Hitting a non-standard centre distance	Common. A housing whose bore centres are already fixed, or a standard centre distance that the tooth counts will not give	EITHER SIGN, split between the two gears. The centre distance is set by the TOTAL; how it is split is a free choice and should be spent on the pinion	A positive total raises the operating pressure angle and shortens both tips; a negative total lowers it and reduces the contact ratio, which is the direction that gets noisy
3. Balancing specific sliding between pinion and wheel	Used in designed gearing rather than in stock gearing. A small pinion’s root slides far more per mesh than the wheel’s does, which is where scuffing and pitting start	POSITIVE on the pinion, negative on the wheel, summing to zero. That is the S0 pair, and the centre distance does not move	Nothing is free: the wheel’s tip thins and the wheel’s root gets shallower. This page does NOT compute specific sliding — see the note below the table
4. Balancing bending strength between pinion and wheel	Used in designed gearing. The pinion meshes z₂/z₁ times as often as the wheel, so equal tooth stresses mean unequal lives; equal LIVES need the pinion stronger	POSITIVE on the pinion. It thickens the root, which is where the bending stress is	The pinion’s tip thins, and the tip is where a case-hardened tooth is most brittle. The two effects fight, which is why the tip thickness has to be checked every time
5. Avoiding a narrow top land	The constraint rather than the objective. It is what stops all four of the above	NEGATIVE, or less positive, on whichever gear is pointing	Everything the positive shift bought goes back

How much positive shift a tooth will take before it points, at 20°

Read the last column. It is the whole of the small-pinion problem in one number: the window between the shift a gear NEEDS to clear undercut and the shift at which its tip points. At 20 teeth there is no undercut to clear at all and the window is the whole positive range up to 1.22. At 12 teeth it has closed to 0.52, and at 10 teeth to 0.28 — and that is measured against a POINTED tip, not against a usable one. Hold a 0.25·m top land instead and the 10-tooth pinion has only 0.029 of shift to play with, which is why ten teeth is close to the practical floor for a 20° spur pinion and why designers who need fewer reach for a 25° pressure angle (which halves the undercut limit to 11.2 teeth) or for helical teeth (whose undercut limit is on the VIRTUAL tooth count, z/cos³β, and is therefore always easier). The undercut column is repeated from the spur gear geometry page, which owns it; it is here only so the window can be subtracted. One caveat that makes this table CONSERVATIVE: the tip thicknesses here are for an unshortened tip, d_a = m(z + 2 + 2x). In a real pair the tip is shortened by Δy·m, which moves the measurement to a smaller radius and makes the top land a little THICKER than the table says. 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.
Teeth z	Tip thickness unshifted (× m)	x to clear undercut	x at a 0.40·m tip	x at 0.25·m	x at 0.20·m	x at a POINTED tip	Usable window
10	0.5877	0.4151	0.267	0.444	0.499	0.700	0.285
12	0.6209	0.2981	0.350	0.543	0.602	0.820	0.522
15	0.6564	0.1227	0.464	0.677	0.742	0.983	0.861
17	0.6741	0.0057	0.533	0.759	0.828	1.083	1.078
20	0.6949	0.0000	0.631	0.874	0.948	1.222	1.222
25	0.7198	0.0000	0.779	1.047	1.129	1.433	1.433
30	0.7374	0.0000	0.914	1.204	1.294	1.622	1.622
40	0.7607	0.0000	1.153	1.483	1.585	1.959	1.959
60	0.7857	0.0000	1.555	1.952	2.073	2.522	2.522
100	0.8072	0.0000	2.194	2.694	2.847	3.000	3.000

Three coefficients, one inequality, and the tip that stops you

A profile-shifted tooth is not a different curve. It is a different part of the same curve. The involute is generated by the base circle and by nothing else, and the base circle does not move when the tool does. Withdrawing the cutter by x·m simply means the tooth uses a portion of that involute further out from the base circle: fatter at the root, thinner at the tip, more steeply inclined. That is why a shifted gear still meshes conjugately with an unshifted one of the same module and pressure angle — the conjugate action is a property of the involute, not of where the tooth sits on it. What changes is the centre distance at which they mesh without backlash, and everything on this page follows from that.

The centre distance does not grow by the sum of the shifts, and this is the single most common error on the subject. The sum of the shifts adds tooth thickness. The centre distance has to open enough to make room for that extra thickness — but opening a centre distance also raises the operating pressure angle, and a higher pressure angle is more efficient at absorbing thickness. So the opening needed is LESS than the thickness added, by an amount that depends on the tooth counts. The formal statement is that the centre distance grows by y·m where inv α_w = inv α + 2(x₁+x₂)tan α/(z₁+z₂) and a_w = a·cos α/cos α_w, and y is always below x₁+x₂ for a positive total. The leftover, Δy = (x₁+x₂) − y, is not an error term. It is a real length: it is exactly how much each tip circle has to shrink to stop the tips fouling the mating roots.

Where the tip shortening comes from, in one line. Take the bottom clearance between gear 1’s tip and gear 2’s root at the operating centre distance: c = a_w − d_a1/2 − d_f2/2. Substitute a_w = a + y·m, d_a1 = m(z₁+2+2x₁) and d_f2 = m(z₂−2.5+2x₂), and everything with a tooth count in it cancels. What is left is c = m(0.25 − (x₁+x₂−y)). The basic rack asks for 0.25·m of clearance; the shift has eaten Δy·m of it; so Δy·m has to come off the tip radius to give it back. No standard has to tell you that and no table is needed for it — it is arithmetic, and it is why this page computes rather than looks up.

Positive shift helps the root and hurts the tip, and that is the whole trade. A positive shift thickens the tooth at the reference circle by 2x·m·tan α, which lands where the bending stress is highest and is worth real fatigue life. It moves the contact away from the region of highest specific sliding near the root. It clears undercut. Against all of that it thins the top land, and it keeps thinning it until the two flanks meet and the tooth points. The table above gives the shift at which that happens for each tooth count, and the window between clearing undercut and pointing is what makes a very small pinion hard rather than impossible. A case-hardened gear closes the window further, because a thin top land lets the case fronts from the two flanks meet, and a through-hardened tip is brittle and chips.

What this page does and does not own. It does NOT compute the undercut limit or the minimum shift that clears it — the spur gear geometry page does, and duplicating a calculation is how two pages come to disagree. It does not compute the contact ratio, which a shifted pair changes in both directions at once (a higher operating pressure angle shortens the path of contact, shortened tips shorten it further, and a larger centre distance lengthens the base pitch); the contact ratio and backlash page takes the operating centre distance and the shortened tip diameters this page produces and finishes the job. It does not compute specific sliding or flash temperature, for the reasons given under the second table. And it is a geometry page: it says what the teeth will be, not what they will carry. The bending stress page is the next step, and a rating to ISO 6336 or AGMA 2001 is beyond anything on this site.

Frequently asked questions

Does the centre distance increase by (x₁+x₂)·m?

No, and this is the commonest mistake about profile shift. It increases by y·m, where y is the centre distance modification coefficient and is always SMALLER than the sum of the shifts when that sum is positive. For a module 2, 15/30 pair with a total shift of 0.7, the sum would say 1.400 mm and the truth is 1.275 mm. The reason is that opening the centre distance also raises the operating pressure angle, and the higher angle absorbs the added tooth thickness more efficiently. The 0.125 mm difference is not lost: it is the tip shortening, and it has to come off the tips or they foul the mating roots.

What is the tip shortening coefficient?

Δy = (x₁+x₂) − y: the amount, in modules, that each gear’s tip RADIUS has to be reduced so that the bottom clearance stays at the basic rack’s 0.25·m. It falls straight out of the clearance equation, which reduces to c = m(0.25 − Δy) with every tooth count cancelling. The German literature calls it Kopfkürzung and some texts write it k. It is always zero or positive, it is zero exactly when the total shift is zero, and forgetting it is what makes a shifted pair bottom out.

How much profile shift can I put on a pinion?

Until the tip points, and in practice rather less. A 20-tooth 20° pinion can take about 1.22 before the top land goes to zero and about 0.87 before it reaches 0.25 of a module; a 12-tooth one about 0.82 and 0.54; a 10-tooth one about 0.70 and 0.44. Subtract the shift needed to clear undercut — 1 − z/17.0973 at 20° — and what is left is the usable window. The table on this page prints all of it. If the window is too small, the answer is a larger pressure angle (25° drops the undercut limit to 11.2 teeth) or helical teeth, not more shift.

Why does the operating pressure angle change at all?

Because the gears no longer touch on their reference circles. A shifted pair meshes at a larger centre distance, so the circles that actually roll on each other — the OPERATING pitch circles — are larger than the reference circles, and the line of action makes a steeper angle with their common tangent. The base circles have not moved, which is why the involute is still conjugate; only the part of it in contact has. Practically it matters because the separating force on the shafts goes as tan α_w, and a pair running at 26° puts about 34 per cent more radial load into the bearings than a standard one for the same torque — tan 26° over tan 20°.

What is an S0 gear pair?

One where the two shifts are equal and opposite, so the total is zero. Everything then simplifies: the operating pressure angle equals the cutting pressure angle, y and Δy are both zero, there is no tip shortening, and the centre distance is the standard m(z₁+z₂)/2 — so the pair drops into a standard housing. What has changed is that the pinion’s root is thicker and its tip thinner, and the wheel’s the reverse. It is the cheapest way there is to make the pinion stronger, and since the pinion meshes z₂/z₁ times as often as the wheel, the pinion is where the fatigue life is decided.

Can I mesh a profile-shifted gear with a standard one?

Yes, and it is done constantly — a shifted pinion running against an unshifted wheel is just a pair with x₂ = 0. The involute is conjugate regardless of shift, so the action is correct. What you must NOT do is assume the standard centre distance: a total shift of x₁ alone still moves the operating centre distance by y·m and still needs the tips shortened by Δy·m. A shifted pinion dropped into a standard-centre housing runs with backlash, which is sometimes exactly what is wanted and is never an accident worth having.

Is the minimum tooth tip thickness 0.25 of a module?

That is the figure in general circulation and this batch could not trace it to a named publication, which is why it is an input on this page rather than a constant. The one value that could be verified in print is 0.2 m_n, from Gear Technology’s Nonstandard Tooth Proportions, with the reason: a hard pointed tooth is brittle. For a carburised gear, stricter still is defensible, because the case grows in from both flanks and from the top land, and once they meet the tip is through-hardened and unsupported. Use your heat treater’s figure if you have one; use 0.2 as the floor if you do not.

Related calculators

References

ISO 21771:2007, Gears — Cylindrical involute gears and gear pairs — Concepts and geometry. Cited by number and not reproduced. It is the document that fixes the symbols used on the profile shift page — x for the profile shift coefficient, α_wt for the working transverse pressure angle, a_w for the working centre distance — and that separates the sum of the shifts from the centre distance change, which is the whole subject of that page.
DIN 3992, Addendum modification of external spur and helical gears (Profilverschiebung bei Stirnrädern mit Aussenverzahnung), and ANSI/AGMA 901-A92, A Rational Procedure for the Preliminary Design of Minimum Volume Gears, Annex A. Both cited by number and neither reproduced. DIN 3992 is the origin of the German term for the quantity this page calls tip shortening — Kopfkürzung — and AGMA 901-A92 Annex A is the published source for the complete shifted-pair equations. The equations on this page were DERIVED from the basic rack rather than copied from either, and then checked against a third, independent statement of them (below).
tec-science, Calculation of involute gears and Profile shift of involute gears (read 29 September 2026). Used as the independent check on the derivation, not as its source. It states the operating pressure angle as inv α_b = 2(x₁+x₂)tan α₀/(z₁+z₂) + inv α₀, and the shortened tip diameter in the equivalent form d_a1* = 2a − m(z₂ + 2x₂ − 2). That form and the one derived here from the 0.25 m clearance agree to machine precision at every parameter set tried, which is the check worth having: two different routes to the same number.
Nonstandard Tooth Proportions, Gear Technology (geartechnology.com, read 29 September 2026). The source for the only minimum top-land figure this batch could verify in print: a tip thickness “minimum of 0.2 m_n”, with the reason — a hard pointed tooth is brittle — and the statement that profile shift cannot be pushed past the pointed-tooth condition. The widely repeated 0.25 m could NOT be traced to a named publication in this batch, which is why the limit on this page is an input and not a constant.
Profile Shift, Gear Technology (read 29 September 2026). The source for the ORDER in which profile shift is used, which it takes from ANSI/AGMA 901-A92: avoiding undercut, balanced specific sliding, balanced flash temperature, balanced bending fatigue life, and avoiding narrow top lands. Also the source for the history — the idea dates from the last quarter of the nineteenth century and began as an undercut fix before anyone noticed what else it did.
Stock Drive Products / Sterling Instrument, Elements of Metric Gear Technology (the technical section of catalogue D805), section 4 on profile shifted gears. Cited for the framing this batch takes from it — that in a shifted pair it is the OPERATING pitch circles that roll on each other, not the reference circles, which is why the operating pressure angle is a different number from the cutting pressure angle and why almost every consequence of a profile shift follows from that one fact. Its own tables 4-4 and 4-5 are published as images and were not transcribed.
ISO 53:1998, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile. Cited by number. It is the document that makes a rack the reference object rather than merely another gear: the basic rack IS the profile, and every involute gear in the system is defined as what that straight-sided rack generates. What is taken from it here is four coefficients, not a table — addendum 1 m, clearance 0.25 m, hence dedendum 1.25 m and whole depth 2.25 m, with a 20° flank angle. DIN 867 is the same profile.