Gear Contact Ratio and Backlash Calculator

Transverse contact ratio from the tip and base circles and the line of action, banded against published guidance; and the backlash a centre-distance offset and a tooth thinning produce — exact, not linearised — with the catalogue band and the thermal change.

Last updated 24 Sep 2026

Contact ratio and backlash

Teeth, centre distance and thinning → mesh and backlash

Module m (mm)

Teeth on the pinion z₁The contact ratio is dominated by the SMALLER gear, because it is the pinion’s tip circle that limits the path of contact at one end.

Teeth on the wheel z₂

Pressure angle αA LOWER pressure angle gives a HIGHER contact ratio, which is the one thing 14.5° is better at than 20°: 1.92 against 1.64 for the same teeth. It also makes the backlash more sensitive to centre distance, because both go as tan α.

Centre distance increase Δa above nominal (mm)How much wider apart the shafts are than m(z₁+z₂)/2. This is where most of your backlash comes from, and it is set by the housing’s bore centres and their tolerance — see the tolerance stack-up calculator.

Tooth thinning per gear, from cutting deeper (mm)Backlash deliberately cut into the teeth by sinking the hob slightly deeper. Each gear contributes its own thinning, so the pair’s circumferential backlash gets twice this. Stock gears are usually supplied with backlash cut in this way, so that they work at the nominal centre distance.

Materials, for the thermal effectDifferential expansion between the gears and the housing changes the centre distance, and therefore the backlash. A steel gear in an aluminium housing OPENS the backlash as it warms; a cast-iron housing closes it slightly.

Temperature rise above the assembly temperature (K)A gearbox at 80 °C assembled at 20 °C has a 60 K rise. Note that the housing and the gears may not be at the same temperature in service, which this page cannot know — the gears run hotter.

Not a circuit: a construction of the mesh, not a scale drawing of it. The two arcs are the base circles, the straight line between them is the line of action — the common tangent to both — and the ticks along it are one base pitch apart, because that is the spacing at which successive tooth pairs arrive on it. The contact ratio is the length of the path of contact measured in those base pitches, and the bar underneath draws it on exactly that scale, quantised to a fortieth. The vertical line at 1.0 is the one that matters: a bar shorter than it means the leaving tooth pair lets go before the next one picks up, so there is a moment in every mesh cycle with no contact at all and the pair hammers instead of rolling. The published floor is 1.1 at worst case and good practice is 1.2, so a bar that only just clears the line is not a pass.

1.5860Example

A 2 mm module 20/40 pair at 20°, assembled 0.1 mm wide of nominal, with 0.03 mm of tooth thinning per gear, steel gears in an aluminium housing running 40 K above assembly

The path of contact over the base pitch, and two sources of backlash

ε_α = [√(rₐ₁² − r_b₁²) + √(rₐ₂² − r_b₂²) − a’·sin α_w] / (πm·cos α) · cos α_w = a·cos α/a’ · j_t = 2a'(inv α_w − inv α) ≈ 2·Δa·tan α · j_n = j_t·cos α · Δa_thermal = (α_housing − α_gear)·a·ΔT

rₐ, r_b: tip and BASE radii. The base radius, not the root radius — at least one published contact-ratio formula prints the root diameters here, which is wrong, and is why this page derives the path of contact from the line of action and checks it against a construction
α_w: working pressure angle at the actual centre distance a’. Equal to α only when a’ is nominal
πm·cos α: the base pitch: the spacing of successive tooth contacts along the line of action. The contact ratio is the path of contact measured in base pitches
inv: the involute function tan − angle. The exact backlash formula needs it at both the working and the nominal pressure angle
j_t, j_n: circumferential and normal backlash. The circumferential one is the lost motion measured on the pitch circle; the normal one is the gap measured along the tooth normal, and they differ by cos α. Sources disagree about which name goes with 2Δa·tan α, so this page prints both and says which is which

Worked example

A 2 mm module 20/40 pair at 20°, assembled 0.1 mm wide of nominal, with 0.03 mm of tooth thinning per gear, steel gears in an aluminium housing running 40 K above assembly
The contact ratio first, because it is the number that decides whether the pair runs at all. The path of contact is the piece of the line of action between the two tip circles: √(rₐ₁² − r_b₁²) + √(rₐ₂² − r_b₂²) − a'·sin α_w. At the nominal centre distance that is 9.6546 mm, and the base pitch πm·cos α is 5.9043 mm, so ε_α = 1.5860. That means that for 59 per cent of the mesh cycle two tooth pairs share the load and for the rest one pair carries everything
But the pair is assembled 0.1 mm wide of nominal, and the headline figure is the contact ratio at the centre distance it actually runs at. Opening the centres raises the working pressure angle and shortens the path of contact, so ε_α falls to 1.5860 — 0.0492 lost for a tenth of a millimetre. That sensitivity is the other half of the backlash trade below
Band it against the published guidance, which SDP/SI states as “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.” 1.59 is comfortably clear. Below 1.0 the mesh would actually break contact between tooth pairs and the pair would not run; between 1.0 and about 1.2 it runs, noisily, with the load arriving as a shock each time a pair engages
Now the backlash from the 0.1 mm of centre-distance offset. The working pressure angle rises: cos α_w = a·cos α/a', so α_w = 20.2603° rather than 20°, and the exact backlash is 2a'(inv α_w − inv α) = 0.0734 mm. The formula everyone prints, 2·Δa·tan α, gives 0.0728 mm — -0.79 per cent LOW. It is the first-order term, it is always low, and at a millimetre of offset it is out by seven per cent
Add the thinning. Each gear was cut 0.030 mm thin on the tooth, and both contribute, so that is another 0.060 mm of circumferential backlash. Total j_t = 0.1334 mm. As a NORMAL backlash — the gap measured along the tooth normal — that is j_t·cos α = 0.1253 mm, and as lost motion at the pinion it is 22.9 arc minutes
Is that the right amount? KG Stock Gears' published band for a spur pair of this module is 0.04 m to 0.12 m, which is 0.08 to 0.24 mm. 0.133 mm sits inside it. Worth seeing how narrow that window is in housing terms: the whole recommended range corresponds to only 0.110 to 0.330 mm of centre-distance offset, which is why stock gears normally carry their backlash in the TEETH and are designed to run at the nominal centre distance
Finally the temperature. Steel gears (11.7 µm/m·K) in an aluminium housing (23.0) over a 40 K rise: the housing grows more than the gears, so the centre distance grows by (α_h − α_g)·a·ΔT = 27.1 µm and the backlash OPENS by 0.0197 mm — about 25 per cent of the catalogue minimum, in the safe direction. A cast-iron housing (10.5) goes the other way and CLOSES it by 0.00210 mm, which is small here but scales with the centre distance and the temperature rise. The case that actually bites is a steel pair in a steel housing where the GEARS are hotter than the case, which this page cannot know and which the thermal effect on fit calculator is the place to think about

Transverse contact ratio, by tooth count and pressure angle

Two things to read off this. The contact ratio rises with tooth count — slowly, and it is dominated by the SMALLER gear, because it is the pinion’s tip circle that limits one end of the path of contact. And a LOWER pressure angle gives a HIGHER contact ratio, which is the one respect in which the legacy 14.5° form beats 20°: 1.92 against 1.64 on a 20/40 pair. That is why 14.5° survived in instrument and clock gearing, where quietness and smooth transmission matter more than tooth strength. SDP/SI’s guidance is the standard one and is worth quoting exactly: “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.” Note the last clause — the number that matters is the worst-case one, not the nominal one, so a nominal 1.25 with loose tolerances is not safe.
Teeth z₁ / z₂	At 20°	At 14.5°	At 25°
12 / 12	1.420	1.581	1.315
14 / 28	1.550	1.782	1.405
17 / 17	1.515	1.722	1.382
20 / 40	1.635	1.920	1.461
25 / 50	1.683	2.002	1.491
30 / 30	1.654	1.947	1.474
40 / 80	1.770	2.160	1.545
60 / 120	1.828	2.276	1.579
100 / 100	1.853	2.324	1.593

Recommended circumferential backlash, and the centre-distance offset that produces it

The three middle columns are KG Stock Gears’ own published band, from their technical data section 1.7: for one spur pair of the same material in the 0.9 to 3 mm module range, 0.04 m to 0.10 m for steel and 0.06 m to 0.12 m for stainless, brass or plastic, with 0.06 m to 0.12 m for steel from 3 to 5 mm. It is a CATALOGUE value, not a standard value, and it is what a stock-gear maker will actually hold — a backlash specification is meaningless without an accuracy grade behind it (ISO 1328-1), because the gears’ own tooth-to-tooth and total composite errors eat into the budget before the centre distance is considered. The last two columns turn the band into the number a housing designer can use: on a 2 mm module 20° pair, the whole recommended backlash range corresponds to just 0.11 to 0.33 mm of centre-distance offset. That is a tight window for a bored housing, which is why most stock gears have backlash cut into the TEETH instead and are designed to run at the nominal centre distance. These dimensions come from a published standard’s table, not from a formula. The standard itself is cited below and the printed values are attributed to the catalogue they were taken from; a different publisher may round differently in the last digit.
Module (mm)	0.04 m (mm)	0.10 m (mm)	0.12 m (mm)	Δa for 0.04 m (mm)	Δa for 0.12 m (mm)
0.50	0.020	0.050	0.060	0.0275	0.0824
1.00	0.040	0.100	0.120	0.0549	0.1648
1.50	0.060	0.150	0.180	0.0824	0.2473
2.00	0.080	0.200	0.240	0.1099	0.3297
3.00	0.120	0.300	0.360	0.1648	0.4945
4.00	0.160	0.400	0.480	0.2198	0.6594
5.00	0.200	0.500	0.600	0.2747	0.8242

2·Δa·tan α is a linearisation, and it is low

For a 2 mm module 20/40 pair. The formula everybody prints, j_t = 2·Δa·tan α, is the first-order term of the exact involute answer j_t = 2a'(inv α_w − inv α), where cos α_w = a·cos α/a’. It is low, always, and by more than you would guess: 0.8 per cent at a tenth of a millimetre, 1.6 per cent at two tenths, 7.2 per cent at a millimetre, and roughly twice those figures at 14.5°. For ordinary housing tolerances the linearisation is fine and this page shows both; for a large deliberate centre-distance offset, or for a minimum-backlash calculation where the error eats the margin, use the exact form. Note also which way the error goes: the linearisation UNDERSTATES the backlash you will get, so designing minimum backlash with it is the safe direction and designing maximum backlash with it is not.
Δa (mm)	Exact backlash (mm)	2·Δa·tan α (mm)	Error at 20°	Error at 14.5°
0.02	0.01458	0.01456	-0.16%	-0.28%
0.05	0.03654	0.03640	-0.40%	-0.70%
0.10	0.07337	0.07279	-0.79%	-1.38%
0.20	0.14789	0.14559	-1.56%	-2.71%
0.50	0.37822	0.36397	-3.77%	-6.39%
1.00	0.78414	0.72794	-7.17%	-11.71%

Why zero backlash is not the goal

Too little backlash is a failure mode and not merely a compromise. A mesh that closes up jams: the two flanks of each tooth contact at once, the friction rises sharply, the heat rises with it, the parts grow, and the backlash closes further. The end of that sequence is a seized or scuffed pair, and it happens on warm-up rather than at rated load, which is why it is missed in testing. Zero-backlash arrangements do exist — split scissor gears with a spring between the halves, and preloaded anti-backlash assemblies — but they achieve it by ELASTICALLY taking up the clearance rather than by removing it, so the clearance is still there for the oil and the expansion. If positioning accuracy is what you need, that is the mechanism to use; do not get it by boring the housing close. 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.
What backlash is for	What happens without it
Thermal expansion	Gears and housing grow by different amounts, and the gears run hotter than the housing does. A steel pair in a cast-iron housing over a 60 K rise loses about 0.03 mm of backlash on a 60 mm centre distance; a hot spot or a warm-up transient can do worse
The lubricant film	The teeth have to have somewhere for the oil to go. With zero backlash the trailing flank of each tooth is pressed against its mate and the film on that side is squeezed out, which raises the friction, raises the heat, and closes the backlash further — a loop that ends in scuffing
Tolerances on everything	The gears’ own tooth-to-tooth and total composite errors, the runout of each blank, the bore-to-bore distance of the housing, the fit of each gear on its shaft and the shaft’s own deflection under load all add up. Every one of them can close the mesh at some angular position even when the average is right
Deflection under load	The shafts bend, the teeth themselves deflect, and the housing spreads. Under load the working centre distance is not the assembled one, and the teeth are not parallel to each other
Contamination and debris	A backlash of a few hundredths of a millimetre is also the clearance that lets a wear particle pass through the mesh rather than being rolled into a tooth flank

Below 1.0 it will not run, and zero backlash is not the goal

The contact ratio is the path of contact measured in base pitches, and below 1.0 the gear will not run. The path of contact is the part of the line of action that lies inside both tip circles: √(rₐ₁² − r_b₁²) + √(rₐ₂² − r_b₂²) − a’·sin α_w, and successive tooth pairs arrive on it one base pitch apart, so dividing by πm·cos α gives the average number of pairs in contact. Note the BASE radii — at least one published version of this formula prints the root diameters, which is wrong, and the version here was derived from the line of action and checked against a direct geometric construction of the path of contact at eight parameter sets. Below 1.0 the mesh breaks contact between pairs, the drive stops being continuous and the gears hammer. SDP/SI’s guidance is worth quoting exactly: “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.”

Backlash comes from two places and they are independent. Opening the centre distance gives j_t = 2a'(inv α_w − inv α), whose first-order term is the familiar 2·Δa·tan α; that linearisation is LOW, always, by 0.8 per cent at a tenth of a millimetre and 7.2 per cent at a millimetre, and roughly twice as much at 14.5°. Cutting the teeth thinner gives backlash directly, and both gears contribute, which is how most stock gears are supplied — so that they work at the NOMINAL centre distance. The two add. And note that the circumferential backlash j_t and the normal backlash j_n differ by cos α, that sources disagree about which name goes with 2Δa·tan α, and that this page therefore prints both and says which is which.

Zero backlash is not the goal. A gear pair needs backlash for four separate reasons: differential thermal expansion, the lubricant film, the accumulated tolerances of the gears, the shafts and the housing, and deflection under load. Take it away and the mesh contacts on both flanks of every tooth at once: friction rises sharply, heat rises with it, both parts grow, and the backlash closes further. That loop ends in a scuffed or seized pair, and it happens during warm-up rather than at rated load, which is exactly why it gets missed. Where positional accuracy really is needed, the answer is a split scissor gear or a preloaded anti-backlash assembly, which take the clearance up ELASTICALLY and leave it there for the oil — not a housing bored close.

How much, then. KG Stock Gears’ published band for a spur pair is 0.04 m to 0.12 m of circumferential backlash, depending on material and module band — a catalogue figure rather than a standard one, and it is what a stock-gear maker will actually hold. The thing to notice is how narrow that is in housing terms: on a 2 mm module 20° pair the whole range corresponds to 0.11 to 0.33 mm of centre-distance offset. A backlash specification also means nothing without an accuracy grade behind it, because the gears’ own tooth-to-tooth and total composite errors (ISO 1328-1) consume part of the budget before the centre distance is considered.

Temperature closes it, and the direction depends on the housing. Differential expansion changes the centre distance by (α_housing − α_gear)·a·ΔT, and the backlash follows it. Steel gears in an aluminium housing OPEN as they warm — aluminium grows twice as fast — which is the safe direction. Steel gears in a cast-iron housing CLOSE, because iron grows slightly less than steel. The case that actually bites is the one this page cannot compute: the gears run hotter than the case, because the mesh is where the loss appears and the case is what sheds it, so even a matched pair of materials closes up on warm-up. Design the cold backlash so the HOT backlash is in band. For the expansion arithmetic on a fit rather than a mesh, see the thermal effect on fit calculator, and for the centre-distance tolerance itself the tolerance stack-up calculator.

Frequently asked questions

What is a good contact ratio for a spur gear pair?

1.2 or more, and never below 1.1 at worst case. SDP/SI’s Elements of Metric Gear Technology puts it as “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.” The last clause is the part people skip: the 1.1 applies after the tolerances have been taken to their limits, so a nominal 1.15 is already there. A typical 20° pair of ordinary tooth counts sits between 1.5 and 1.8, and more is better for noise.

What happens if the contact ratio is below 1?

The pair does not work. Below one, the leaving tooth pair breaks contact before the next pair picks up, so there is a moment in every mesh cycle with no contact at all: the drive is discontinuous, the gears hammer instead of rolling, and both the noise and the impact load are unbounded. It is not a degraded condition but a non-functioning one. The usual causes are too few teeth on the smaller gear or a centre distance far wider than nominal.

How much backlash should a gear pair have?

KG Stock Gears’ published band for one spur pair of the same material is 0.04 m to 0.10 m of circumferential backlash for steel in the 0.9 to 3 mm module range, and 0.06 m to 0.12 m for stainless, brass, plastic or for steel from 3 to 5 mm. So about 0.08 to 0.24 mm at module 2. Treat it as a catalogue value rather than a standard one, and note that a backlash figure is meaningless without an ISO 1328-1 accuracy grade behind it, because the gears’ own errors eat into the budget first.

How does centre distance affect backlash?

j_t = 2a'(inv α_w − inv α) exactly, where cos α_w = a·cos α/a’; to first order that is the familiar 2·Δa·tan α. On a 20° pair, a tenth of a millimetre of extra centre distance gives about 0.073 mm of circumferential backlash. The linearisation is always LOW — 0.8 per cent at 0.1 mm and 7.2 per cent at 1 mm, and about twice that at 14.5° — so it is the safe formula for setting a minimum backlash and the unsafe one for checking a maximum.

Why can’t a gear pair have zero backlash?

Because four separate things need the clearance: differential thermal expansion between the gears and the housing, the lubricant film on the non-driving flank, every tolerance in the chain from tooth spacing to housing bore centres, and deflection under load. Without it the mesh contacts on both flanks at once, friction and heat rise, the parts grow, and the backlash closes further until the pair scuffs or seizes — usually on warm-up rather than at load. Where zero lost motion is genuinely needed, a split scissor gear or a preloaded assembly takes the clearance up elastically and leaves it there for the oil.

Does temperature change backlash?

Yes, through differential expansion of the centre distance: Δa = (α_housing − α_gear)·a·ΔT, and the backlash changes by about twice that times tan α. Steel gears in an aluminium housing open up as they warm, because aluminium grows about twice as fast as steel; steel gears in a cast-iron housing close slightly. The case that causes trouble is the one no simple calculation catches: the gears run hotter than the housing, because the mesh generates the loss and the housing sheds it, so even matched materials close up during warm-up. Design the COLD backlash so the hot one is in band.

What is the difference between circumferential and normal backlash?

Circumferential backlash j_t is the lost motion measured along the pitch circle — how far one gear turns before the other picks up. Normal backlash j_n is the gap between the flanks measured perpendicular to the tooth surface, which is what a feeler gauge or a shim in the mesh measures. They differ by cos α: j_n = j_t·cos α. Radial backlash is a third quantity, j_t/(2·tan α), and is the centre-distance change that would produce the same lost motion. Published sources are not consistent about which of these 2Δa·tan α gives, so this page prints all three and labels them.

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.
KG Stock Gears (Kohara Gear Industry), Technical Data section 1.7, Backlash. The source for the recommended circumferential backlash band used here: for a spur pair of the same material in the 0.9 to 3 mm module range, 0.04 m to 0.10 m for steel and 0.06 m to 0.12 m for stainless, brass or plastic, rising to 0.06 m to 0.12 m for steel in the 3 to 5 mm range. Quoted as a catalogue value rather than a standard value, which is what it is.
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.
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.
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.