Helical Gear Geometry and Thrust Calculator

Normal against transverse module and pressure angle, centre distance, the axial thrust F_t·tan β that goes into the bearings, and the overlap ratio that makes a helical pair quiet — with the bearing types that can and cannot take the thrust.

Last updated 24 Sep 2026

Helical gear geometry and thrust

Normal module, teeth, helix angle → geometry and thrust

Normal module m_n (mm)The module measured PERPENDICULAR to the tooth, which is the one a hob cuts and the one a catalogue quotes. It is not the module the pitch diameter uses.

Teeth on the pinion z₁

Teeth on the wheel z₂

Helix angle β (degrees)Measured from the gear AXIS. 8° to 25° covers almost all single-helical gearing; 30° to 45° is double-helical territory because the thrust becomes unmanageable otherwise. Thrust is F_t·tan β, so 20° costs you 36 per cent of the tangential force into the bearings.

Normal pressure angle α_nMeasured in the normal plane. The TRANSVERSE pressure angle, which is the one the contact geometry actually uses, is larger: tan α_t = tan α_n/cos β.

ArrangementA double helical (herringbone) pair has two opposite helices, so the two thrusts cancel inside the gear and none reaches the bearings — at the cost of needing one shaft axially free to let the halves find their own centre.

Face width b (mm)Sets the overlap ratio, which is what makes a helical gear quiet: ε_β = b·sin β/(π·m_n). It also sets the thrust bearing’s job, because a wide face needs the load spread across it and a misaligned wide face concentrates it.

Power transmitted (kW)

Pinion speed (rev/min)

Not a circuit: the pitch cylinder unrolled flat, so the tooth traces become straight lines at the real helix angle, measured from the axial direction as gearing measures it. Three traces are drawn one transverse pitch apart. The triangle on the right is the force split, drawn TO SCALE: the horizontal arrow is the tangential force, the vertical one is the axial thrust, and the ratio between them is tan β — so at 20° the thrust arrow is 36 per cent of the tangential one, and at 45° it is the same length. That is the whole cost of a helix in one picture. Both the traces and the triangle are quantised to about a fortieth of the helix-angle range. Selecting a double helical arrangement leaves the thrust arrow drawn — because each half still generates it internally and the gear's own rim has to carry it — and adds a note that the bearings see none of it.

979.3NExample

A 3 mm normal module 23/57 helical pair at a 20° helix and a 20° normal pressure angle, 40 mm face, carrying 15 kW at 1,450 rev/min

Normal, transverse, and the thrust that comes out of the difference

m_t = m_n / cos β · d = m_t·z · a = m_n(z₁ + z₂)/(2·cos β) · tan α_t = tan α_n / cos β · F_a = F_t·tan β · F_r = F_t·tan α_t · ε_β = b·sin β/(πm_n) · ε_γ = ε_α + ε_β

m_n: normal module — measured perpendicular to the tooth. What the hob cuts and what the catalogue says
m_t: transverse module — measured in the plane of rotation. LARGER than m_n, and the one the pitch diameter uses. Getting these two the wrong way round is the classic helical error
β: helix angle, measured from the gear axis. Everything good about a helix grows with it, and so does the thrust
α_n, α_t: normal and transverse pressure angles. α_t is the larger. Note the direction of the relation: tan α_t = tan α_n/cos β, and at least one published source prints it inverted
F_t, F_a, F_r: tangential, axial and radial tooth forces. The three are the components of one normal force, and the ratios tan β and tan α_t follow from the tooth surface’s own normal direction
ε_β: overlap or axial contact ratio: the face width divided by the axial pitch. It is what a spur gear does not have

Worked example

A 3 mm normal module 23/57 helical pair at a 20° helix and a 20° normal pressure angle, 40 mm face, carrying 15 kW at 1,450 rev/min
Start with the module, because everything downstream depends on getting it the right way round. The hob cut a 3 mm NORMAL module. The transverse module is m_n/cos β = 3/cos 20° = 3.1925 mm, and it is the transverse module that the pitch diameter uses: d₁ = m_t·z₁ = 73.428 mm, not the 69 mm you would get from the normal module. That six per cent is the error that ships
So the centre distance is m_n(z₁ + z₂)/(2·cos β) = 127.701 mm, against 120 mm for the same teeth as a spur pair. The helix has bought you 7.701 mm, and that is the reason helical gears exist as much as noise is: a spur pair's centre distance is fixed by m(z₁+z₂)/2 with no freedom in it at all, while a helix angle is a continuous parameter you can choose to hit whatever the housing gives you
The transverse pressure angle: tan α_t = tan 20°/cos 20°, so α_t = 21.173°. Note which way round that is — the transverse angle is the LARGER of the two. It is the angle the contact geometry uses, so it is the one that sets the base diameter (d·cos α_t = 68.472 mm) and the separating force
Now the loads. 15 kW at 1,450 rev/min is 98.79 N·m at the pinion, which at a 73.4 mm pitch diameter is a tangential force of 2000T/d = 2,691 N. The pitch line velocity is 5.57 m/s
THE THRUST, which is what this page is for: F_a = F_t·tan β = 2,691·tan 20° = 979 N, or 36.4 per cent of the tangential force. That goes straight into the bearings and it does not care what you intended. A pair of plain cylindrical roller bearings, which have the best radial capacity of anything, cannot take a newton of it. The separating force is F_t·tan α_t = 1,042 N, and the resultant in the radial plane — which is what bends the shaft — is 2,885 N
The thrust also applies a moment to the shaft, because it acts at the pitch radius: F_a·d₁/2 = 36.0 N·m of bending, on top of the bending from the radial resultant. On a slender pinion shaft that is often the term that decides the shaft diameter
Finally the quietness, which is the overlap ratio: ε_β = b·sin β/(πm_n) = 40·sin 20°/(π·3) = 1.452. The transverse contact ratio is 1.632, so the total is 3.083 — comfortably over 2, which means there are always at least two tooth pairs in contact and the load transfer is continuous. A spur pair of the same teeth would have only the 1.684 and would step

Normal against transverse, for a 3 mm normal module 23/57 pair with a 40 mm face

Every column here changes with the helix angle and the normal module does not — that is the whole point, and the source of the commonest helical error. The hob cuts the NORMAL module, the catalogue quotes the NORMAL module, and the pitch diameter uses the TRANSVERSE one, m_n/cos β. Use m_n in the pitch diameter and you get a gear that is too small by a factor of cos β — six per cent at 20°, which on this pair is eight millimetres of centre distance. The second column is also the answer to “how do I hit a centre distance the tooth counts will not give me”: unlike a spur pair, a helical pair has a continuous free parameter, and a helix angle can be chosen to any number of decimal places. The cost is in the fifth column.
Helix angle β	Transverse module m_t (mm)	Centre distance (mm)	Transverse pressure angle α_t	Thrust as % of F_t	Overlap ratio ε_β
5.0°	3.0115	120.458	20.070	8.7%	0.370
10.0°	3.0463	121.851	20.284	17.6%	0.737
15.0°	3.1058	124.233	20.647	26.8%	1.098
20.0°	3.1925	127.701	21.173	36.4%	1.452
25.0°	3.3101	132.405	21.880	46.6%	1.794
30.0°	3.4641	138.564	22.796	57.7%	2.122
35.0°	3.6623	146.493	23.957	70.0%	2.434
45.0°	4.2426	169.706	27.236	100.0%	3.001

Where the thrust goes, and which bearing can take it

Helical gearing produces axial thrust and the thrust has to be carried by something. F_a = F_t·tan β means 36 per cent of the tangential force at a 20° helix, which for this page’s default pair is about a kilonewton — not a detail. Two design consequences worth stating plainly. The thrust reverses when the drive reverses, so a reversing drive needs axial capacity in both directions and cannot rely on a single angular-contact row. And the thrust acts at the pitch radius, so it also applies a moment to the shaft equal to F_a·d/2, which is a bending load the shaft deflection and slope calculator can carry — and the slope it produces at the bearing is a bearing life question, not just a stiffness one. This page sizes a part; it does not certify one. Where the answer carries a consequence — a load path, a lifting duty, a pressure boundary, a fastener holding something that can fall — confirm it against the design code that governs the application, and against the manufacturer’s own rating, before relying on it.
Bearing type	Takes axial load?	What that means here
Deep groove ball bearing	Yes, but modestly — roughly a quarter to a half of its radial rating, and only when the radial load is there to keep the balls seated	The default choice for a small single-helical pair at a modest helix angle. Check the combined load, not the axial one alone
Angular contact ball bearing	Yes, in ONE direction, and that is the point of it	A single row takes thrust one way only, so it is fitted against something — a second angular contact bearing, or a deep groove bearing taking the radial load. Pairs are what most single-helical gearboxes use
Tapered roller bearing	Yes, heavily, in one direction	The usual answer when the thrust is large: high capacity in both radial and axial directions, always mounted in opposing pairs, and the pair’s setting decides the shaft’s axial play. A steep helix and a big torque is taper roller territory
Cylindrical roller bearing (without flanges)	NO — none at all	This is the trap. A cylindrical roller bearing has the highest radial capacity of any of these and, in its plain form, cannot take thrust. Fitting one at each end of a helical pinion leaves the thrust with nowhere to go and the shaft walks
Double helical / herringbone gear	The question does not arise	The two opposite helices cancel each other’s thrust inside the gear, so the bearings see none of it. The price is that ONE shaft must be axially free — typically on plain cylindrical rollers — so the two halves can float into balance. Constrain both shafts axially and the halves fight each other

Why a helical gear is quiet: the two contact ratios add

A spur pair (the first row) has only the transverse contact ratio, and at 1.63 it means that for most of the mesh cycle one tooth pair carries everything and the load steps as pairs come in and out. A helical pair adds the overlap ratio, ε_β = b·sin β/(πm_n), which is the face width divided by the axial pitch: contact starts at one end of the tooth and sweeps along it, so teeth engage gradually rather than all at once across the face. Once the total clears about 2 there are always at least two pairs in contact and the load transfer is continuous, which is the actual mechanism behind “helical gears are quieter”. Note that ε_β depends on the FACE WIDTH as well as the helix angle — a narrow helical gear gets very little of the benefit, and below about ε_β = 1 you have paid for the thrust without buying the quietness. The transverse ratio, meanwhile, falls slightly as the helix steepens, because the transverse pressure angle rises.
Helix angle	Transverse ε_α	Overlap ε_β	Total ε_γ	Mesh
0.0°	1.684	0.000	1.684	not continuous
5.0°	1.681	0.370	2.051	continuous
10.0°	1.671	0.737	2.408	continuous
15.0°	1.655	1.098	2.753	continuous
20.0°	1.632	1.452	3.083	continuous
25.0°	1.602	1.794	3.396	continuous
30.0°	1.567	2.122	3.689	continuous

Normal against transverse, the thrust, and why the face width decides the quietness

Normal against transverse is the whole of helical gearing, and getting it backwards is the classic error. A hob cuts the NORMAL module, perpendicular to the tooth, and that is the number a catalogue quotes. The pitch diameter uses the TRANSVERSE module, in the plane of rotation, which is larger: m_t = m_n/cos β. So a 3 mm normal module 23-tooth pinion at a 20° helix has a pitch diameter of 73.43 mm and not 69, and a pair of them sits at a 127.70 mm centre distance and not 120. Six per cent at 20°, and it goes the way that makes your gearbox too small. The same distinction runs through the pressure angle: tan α_t = tan α_n/cos β, with the transverse angle the larger of the two — note the direction, because at least one published source prints that relation inverted.

The helix angle is a free parameter, and that is worth more than it sounds. A spur pair’s centre distance is m(z₁ + z₂)/2 and there is no freedom in it whatever: given the module and the tooth counts, the centre distance is decided. A helical pair’s is m_n(z₁ + z₂)/(2·cos β), and β is continuous. So when a housing hands you a centre distance and a ratio that the tooth counts will not give you, a helix angle solves it exactly — choose β = arccos[m_n(z₁+z₂)/2a] and you land on the number. This is used constantly in gearbox design and it is at least as common a reason for specifying a helical pair as noise is.

What it costs: F_a = F_t·tan β, and the thrust goes straight into the bearings. Eighteen per cent of the tangential force at a 10° helix, thirty-six at 20°, fifty-eight at 30° and a hundred at 45°, where the axial force equals the useful one. That is not a correction, it is a load path, and the bearing has to be chosen for it: a deep groove ball bearing takes some, an angular contact bearing takes a lot in one direction, a tapered roller pair takes a lot in both, and a plain cylindrical roller bearing — which has the highest radial capacity of the four — takes none at all. The thrust also acts at the pitch radius, so it applies a moment of F_a·d/2 to the shaft on top of the bending from the radial force, which the shaft deflection and slope calculator can take further. And it reverses when the drive reverses.

Double helical cancels the thrust and needs an axially free shaft. Two opposite helices on one gear produce equal and opposite axial forces, so the net thrust on the bearings is zero — which is why herringbone gears are used at helix angles no single helical pair could survive. The condition is that the cancellation has to be allowed to happen: one of the two shafts must float axially, usually on plain cylindrical rollers, so the halves find their own balance. If both shafts are axially constrained, every difference between the two helices and every bit of axial misalignment is taken out as tooth load instead of as a small movement. Note also that each half still generates its own thrust internally, so the gear’s rim and web carry it even when the bearings do not.

Why it is quieter: the two contact ratios add. A spur pair has only the transverse contact ratio, and teeth engage across the whole face at once, so the load steps. A helical pair adds the overlap ratio ε_β = b·sin β/(πm_n), which is the face width divided by the axial pitch: contact begins at one end of the tooth and sweeps along it. Total contact ratio is ε_α + ε_β, and above about 2 there are always at least two pairs carrying load, so the transfer is continuous. The thing to notice is that ε_β depends on the FACE WIDTH: a narrow helical gear pays the thrust and gets little of the quietness. For the mesh continuity of the transverse part on its own, and for backlash, see the contact ratio and backlash calculator.

Frequently asked questions

Is the pitch diameter of a helical gear m_n·z or m_t·z?

m_t·z, using the TRANSVERSE module, which is m_n/cos β. This is the single most common helical gear mistake, and it always goes the same way: using the normal module makes the gear come out too small by a factor of cos β, which is six per cent at a 20° helix and thirteen per cent at 30°. The confusion is understandable, because the normal module is what the hob cuts and what the catalogue prints — but the pitch circle lives in the plane of rotation, so it is the transverse module that fills it.

How much axial thrust does a helical gear produce?

F_a = F_t·tan β, where F_t is the tangential force. That is 17.6 per cent of the tangential force at a 10° helix, 36.4 per cent at 20°, 57.7 per cent at 30° and exactly 100 per cent at 45°. It is independent of the pressure angle and of the module. The thrust reverses with the direction of drive, and it acts at the pitch radius, so it also applies a bending moment of F_a·d/2 to the shaft.

Which bearing should carry a helical gear’s thrust?

Something chosen for it. A deep groove ball bearing will take roughly a quarter to a half of its radial rating axially, and only with enough radial load to keep the balls seated. An angular contact bearing takes a great deal in ONE direction, so it is fitted against something. A tapered roller pair takes a great deal in both and is the usual answer for a large thrust. A plain cylindrical roller bearing takes NONE, despite having the best radial capacity of the four — that is the trap. For a reversing drive you need capacity in both directions.

Can I use a helical pair to hit an awkward centre distance?

Yes, and it is one of the main reasons helical gears are specified. Because a = m_n(z₁ + z₂)/(2·cos β) and β is continuous, you can solve for the helix angle that gives exactly the centre distance you have: β = arccos[m_n(z₁+z₂)/2a]. A spur pair cannot do this at all — its centre distance is fixed by the module and the tooth counts. The constraint is that β must land in a sensible range, because the thrust grows with it.

Why are helical gears quieter than spur gears?

Because contact begins at one end of the tooth and sweeps along it rather than arriving across the whole face at once. The measure of it is the overlap ratio, ε_β = b·sin β/(πm_n), which adds to the transverse contact ratio; above a total of about 2 there are always at least two tooth pairs in contact and the load transfer is continuous rather than stepped. The important corollary is that the benefit depends on the FACE WIDTH as well as the helix angle, so a narrow helical gear pays for the thrust without buying much of the quietness.

What is a herringbone gear for?

Cancelling the thrust. Two opposite helices on the same gear produce equal and opposite axial forces, so the bearings see none of the thrust and helix angles of 30° to 45° become usable — which gives a very high overlap ratio and a very quiet, very heavily loaded gear. The condition is that one of the two shafts must be axially FREE, normally on plain cylindrical roller bearings, so the two halves can float into balance. If both shafts are axially fixed, any difference between the two helices is taken out as tooth load.

Does the helix angle change the tooth strength?

Yes, in two opposite directions, and this page does not attempt to net them out. The tooth is longer along its helix than its face width, and the load is shared by more teeth, which helps. But the transverse pressure angle rises with the helix angle, the transverse contact ratio falls slightly, and the load on an individual tooth is applied at an angle to its section. A first bending estimate can be made from the bending stress page using the virtual tooth count z/cos³β, but a real helical rating is an ISO 6336 or AGMA 2001 calculation and is out of scope for this site.

Related calculators

References

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.
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.
ISO 6336 (all parts), Calculation of load capacity of spur and helical gears. Cited and explicitly OUT OF SCOPE for this site. It is named so that the reader knows what a bending number from this page is not: ISO 6336 adds the tooth form factor and stress correction factor for the real root fillet, load distribution across the face, the dynamic factor from a measured accuracy grade, rim thickness, and — the part that usually governs — surface durability.
ANSI/AGMA 2001-D04, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth. Cited and explicitly out of scope, for the same reasons as ISO 6336 and with the same consequence: a Lewis number is a sizing estimate and not a rating.
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.