Bevel Gear Geometry Calculator

Pitch cone angles for any shaft angle, outer and mean cone distances, the face width limit on both published rules with the 11 per cent disagreement shown, the mounting distances a bevel pair is actually assembled to, and what a long face does to the tooth at the small end.

Last updated 24 Sep 2026

Bevel gear geometry

Teeth and shaft angle → cones and the face width limit

Outer transverse module m_et (mm)The module at the LARGE end of the tooth, which is where a bevel gear is dimensioned. The tooth tapers toward the apex, so the module at the small end is smaller in the ratio R_i/Rₑ — which is exactly why the face width is limited.

Teeth on the pinion z₁

Teeth on the wheel z₂

Shaft angle Σ (degrees)90° for the ordinary right-angle pair, where δ₁ = arctan(z₁/z₂). Other shaft angles are perfectly ordinary and the general formula handles them: δ₁ = arctan[sin Σ/(z₂/z₁ + cos Σ)]. Above 90° one of the members becomes an internal-looking cone and the face width should be smaller than the rule gives.

Tooth formStraight bevel teeth are the cheapest and the noisiest. Spiral bevel teeth engage gradually, like a helical gear, and carry more load. Zerol teeth are curved but have zero spiral angle at mid-face, so they produce no more thrust than a straight bevel — and they take a tighter face width limit.

Which published face-width rule?Two published forms, both current, differing by 11 per cent. MITCalc (following ISO 23509) and Shigley both say 0.30 of the cone distance; GlobalSpec’s reference says one third. The module term is the same in both: 10/P_d is 10 m, and “three times the pitch” is 9.42 m.

Face width b you intend to use (mm)Compared with the limit above. Exceeding it is why bevel gears fail at the SMALL end: the tooth there is R_i/Rₑ of the size it is at the large end, so a long face puts a large part of the mesh on a tooth that is much weaker than the one you dimensioned.

Not a circuit: the bevel pair in section, drawn as the two pitch cones that share an apex, with both cone angles at their real computed values and the generators quantised to about a fortieth of the 0–90° range. The vertical line at the right of the pinion cone is the crown, at the outer cone distance. The bar underneath is the face width measured ALONG the cone, drawn as a fraction of that cone distance, with a tick marking the computed limit: when the filled part of the bar passes the tick the face is too long, and the note appears. That is the geometry the limit is about — everything on a bevel tooth scales with distance from the apex, so a face that reaches too far back toward the apex ends on a tooth much smaller than the one at the crown that was dimensioned.

26.833mmExample

A 4 mm outer module 20/40 straight bevel pair on perpendicular shafts, with a 25 mm face width

Cones that roll on each other

δ₁ = arctan[sin Σ / (z₂/z₁ + cos Σ)] · δ₂ = Σ − δ₁ · Rₑ = d/(2·sin δ) · R_m = Rₑ − b/2 · b ≤ min(k·Rₑ, 10 m) · m_inner = m·R_i/Rₑ

Σ: shaft angle. 90° for the ordinary pair, where the formula collapses to δ₁ = arctan(z₁/z₂)
δ₁, δ₂: pitch cone angles. They sum to Σ, and they satisfy sin δ₂/sin δ₁ = z₂/z₁, which is the rolling condition and the check this page runs
Rₑ: outer cone distance, apex to the large end of the tooth. The same number from either member, which is the other check
k: the face-width coefficient: 0.30 by MITCalc and Shigley, or 1/3 by GlobalSpec’s reference. Both are printed
m: outer transverse module, at the LARGE end. Every tooth dimension scales with distance from the apex, so the small end is smaller in the ratio R_i/Rₑ
b: face width, measured along the cone. The dimension this page is really about

Worked example

A 4 mm outer module 20/40 straight bevel pair on perpendicular shafts, with a 25 mm face width
The cone angles first. With a 90° shaft angle the general formula collapses to δ₁ = arctan(z₁/z₂) = arctan(20/40) = 26.5651°, and δ₂ = 90° − δ₁ = 63.4349°. The check that they are right is the rolling condition: sin δ₂/sin δ₁ must equal z₂/z₁, and it does, exactly
The pitch diameters at the large end are m·z as usual: 80 mm and 160 mm. The outer cone distance is d/(2·sin δ), which gives 89.443 mm from the pinion — and the same 89.443 mm from the wheel, which it must, because both cones share an apex. At a right angle it is also just the hypotenuse of the two pitch radii
NOW THE NUMBER THIS PAGE IS FOR. The face width limit is the lesser of the cone-distance rule and the module rule. There are two published cone-distance rules and they disagree: MITCalc (following ISO 23509) and Shigley say 0.30 Rₑ = 26.833 mm, GlobalSpec's reference says Rₑ/3 = 29.814 mm. That is 11.1 per cent apart. The module rule, 10 m = 40 mm, is not binding here. So the limit is 26.833 mm on the tighter reading, and your 25 mm face is 93 per cent of it — inside, on either rule
Why the limit exists, in one number. Everything about a bevel tooth scales with its distance from the apex, so the module at the inner cone distance is m·R_i/Rₑ. With a 25 mm face, R_i = 64.443 mm and the small end's module is 2.8820 mm — 72 per cent of the large end's. At the face width limit it would be exactly 70 per cent; at twice the limit, 40 per cent. A long face puts a large part of the mesh on a tooth much weaker than the one you dimensioned, and THAT is why bevel gears fail at the small end
The mean cone distance, which is where a bevel pair's strength is reckoned, is Rₑ − b/2 = 76.943 mm, giving a mean module of 3.4410 mm. The addendum angle is arctan(hₐ/Rₑ) = 2.5606° and the dedendum angle 3.1996°
The two dimensions that decide whether it works when it is assembled are the mounting distances, from each member's locating face to the common apex: 78.211 mm for the pinion and 36.422 mm for the wheel. Both cones must have their apex at the same point. A bevel pair is not set to a centre distance; it is shimmed to these two numbers and then the pinion is moved axially to set the backlash
For a first strength estimate the equivalent spur gear has z/cos δ = 22.36 teeth for the pinion, which is what a Lewis calculation on the mean module would use. That is an estimate and not a rating — a real bevel rating is an ISO 23509 plus ISO 10300 calculation and is out of scope here

The face width limit has two published forms, 11 per cent apart

For the example pair: m = 4, 20/40 teeth, a 90° shaft angle, so Rₑ = 89.443 mm. Both cone-distance forms are in print and current, and this page lets you pick which one it uses rather than averaging them or picking a winner. The difference matters most on a long cone, where the cone-distance term is the binding one; on a short cone with a coarse module the 10 m term binds instead and the two rules agree. MITCalc adds two qualifications worth carrying: for a shaft angle below 90° a larger face width than the rule gives may be used, and above 90° a smaller one should be. 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.
Rule	Coefficient	For this pair (mm)	Source
0.30 Rₑ	0.3000	26.833	MITCalc, following ISO 23509: “generally, the face width is 30 % of the cone distance R_e2 or 10 m_et, whichever is less”. Shigley gives the same coefficient as 0.3 Aₒ
Rₑ/3	0.3333	29.814	GlobalSpec’s bevel gear reference: “the face width should not normally exceed either one third of the cone distance or three times the pitch”
10 m	10	40.000	Shigley’s 10/P_d. GlobalSpec’s “three times the pitch” is 3πm = 9.42 m, the same rule rounded differently — so unlike the cone-distance term, the two sources agree here
0.25 Rₑ, Zerol only	0.25	22.361	MITCalc again: a Zerol face width “should be multiplied by 0.83 and should not exceed 25 % of the R_e2”

Straight, spiral, Zerol — and where the hypoid boundary is

The first three are all bevel gears: their axes intersect, their pitch surfaces are cones, and every formula on this page applies to all of them. The fourth is not, and this page refuses it rather than approximating it. A hypoid pair’s offset makes the pitch surfaces hyperboloids rather than cones, which means the cone angles, the cone distance and the face width rule here are all simply wrong for one; its pinion is usually larger and has a different spiral angle from its wheel; and the sliding it introduces changes the lubricant specification. ISO 23509 covers bevel and hypoid geometry in one document precisely because the distinction needs stating carefully. If your axes do not intersect, you have a hypoid and you need a hypoid calculation.
Tooth form	Geometry	Spiral angle	What it means
Straight bevel	Teeth straight and radial, pointing at the cone apex	0°	Cheapest to make and to inspect, and the noisiest. Contact arrives along the whole tooth at once, like a spur gear. Recommended below about 300 m/min of pitch line velocity, and the right answer when space, weight and mounting cost matter more than noise
Spiral bevel	Teeth curved and oblique, with a spiral angle at mid-face	typically 35°	Contact starts at one end of the tooth and sweeps along it, exactly as in a helical gear, so the mesh is gradual, quiet and carries more load for the same size. The cost is a thrust component that depends on the spiral hand AND the direction of rotation, so the bearing arrangement has to be worked out for both directions if the drive reverses
Zerol bevel	Teeth curved like a spiral bevel but with ZERO spiral angle at mid-face	0° at mid-face	A curved tooth with a straight bevel’s force system: no more thrust than a straight bevel, and the smooth manufacture and the crowned contact of a spiral. Used where a straight bevel’s noise is unacceptable but a spiral’s thrust reversal is a problem. It carries a tighter face-width limit — 25 per cent of the cone distance, with the computed width multiplied by 0.83
Hypoid — NOT A BEVEL GEAR	Axes OFFSET, so they do not intersect at all	n/a	REFUSED by this page, and the boundary is real rather than terminological. A bevel pair’s axes intersect and its pitch surfaces are cones that roll on each other; a hypoid pair’s axes are offset, its pitch surfaces are hyperboloids, and its pinion and wheel spiral angles are NOT equal. Nothing on this page applies. It also needs a different lubricant: the offset adds lengthwise sliding along the tooth that a bevel mesh does not have, the film can be squeezed out, and the oil is an API GL-5 extreme-pressure grade rather than a bevel gear oil

What actually has to happen when you fit a bevel pair

None of this is exotic and all of it gets skipped. The geometry on the rest of this page tells you what to make; this table is what decides whether it works. Two of the numbers it needs are computed above — the two mounting distances — and the third, the backlash, comes from the contact ratio and backlash calculator, where the catalogue band for a bevel pair is wider than for a spur pair of the same module. 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.
The rule	Why
A bevel pair is a MATCHED SET and is replaced as a set	Spiral and Zerol pairs are lapped together in manufacture and are usually marked with a matched-set number. Replacing one member alone puts a tooth surface that was lapped against one mate against a different one, and the contact pattern — which is what a bevel pair’s load capacity actually depends on — is then nobody’s. Order both
Mounting is to a stated cone-apex distance, not to a centre distance	Each member has a mounting distance from its own locating face to the common apex, and it is on the drawing. Both cones must have their apexes at the SAME point; if either member sits at the wrong axial position the contact moves along the tooth toward the toe or the heel and concentrates there. This page computes both mounting distances
Backlash is set by SHIMMING, and it is set by moving the pinion	The wheel is normally shimmed to put its cone apex on the axis intersection, and then the pinion is shimmed axially to set the backlash. Moving the pinion along its own axis changes the backlash without moving the contact pattern much, which is why it is the one used for the adjustment
The contact pattern is the acceptance test	Marking compound on a few teeth, rotated under a light load, shows where the pair is actually touching. A correct pattern is a patch centred on the tooth, clear of both ends and of the tip and root. Too far toward the small end (the toe) or the large end (the heel) means a mounting distance is wrong; a pattern that runs off the end is a pair that will fail there
The tooth at the small end is the weak one	Everything about the tooth scales with distance from the apex, so a tooth at the inner cone distance has module, height and thickness all reduced in the ratio R_i/Rₑ. At the face width limit that ratio is 0.70; at twice the limit it is 0.40. This is the mechanism the face width rule exists to prevent, and it is why an over-wide bevel gear fails at the small end

Two cones, two published face-width rules, and the tooth at the small end

A bevel pair is two cones that roll on each other, and everything follows from that. The pitch cone angles satisfy sin δ₂/sin δ₁ = z₂/z₁ and sum to the shaft angle, which for the general case gives δ₁ = arctan[sin Σ/(z₂/z₁ + cos Σ)] and collapses at a right angle to the familiar arctan(z₁/z₂). The outer cone distance Rₑ = d/(2·sin δ) comes out the same from either member, because both cones share an apex, and that identity is the check worth running on any bevel calculation. The mean cone distance, Rₑ − b/2, is where a bevel pair’s strength is reckoned.

The face width limit is the most useful number on the page, and it has two published forms. The limit is the lesser of a cone-distance term and a module term. MITCalc, following ISO 23509, and Shigley both give the cone-distance term as 0.30 Rₑ; GlobalSpec’s bevel reference gives one third of the cone distance. Those differ by eleven per cent, both are current, and this page lets you pick rather than averaging them. The module terms agree: Shigley’s 10/P_d is 10 m, and GlobalSpec’s “three times the pitch” is 3πm = 9.42 m — the same rule differently rounded. A Zerol pair takes a tighter limit still, 25 per cent of the cone distance.

Why the limit exists: the tooth at the small end is a smaller tooth. Every dimension of a bevel tooth scales with its distance from the apex, so at the inner cone distance the module is m·R_i/Rₑ. At the 0.30 face-width limit that is exactly 70 per cent of the module you dimensioned; at twice the limit it is 40 per cent. The mesh loads the whole face regardless, so an over-wide bevel gear puts a large share of the load on a tooth much weaker than the one the calculation assumed, and it breaks at the toe. There is a second mechanism working the same way: a long thin small-end tooth is less stiff, so mounting error concentrates the contact there instead of spreading it.

Straight, spiral and Zerol — and hypoid gears are not bevel gears. Straight bevel teeth are cheapest and noisiest and are recommended below about 300 m/min. Spiral bevel teeth engage gradually along the tooth like a helical gear, carry more load and are quieter, at the cost of a thrust that depends on the spiral hand and the direction of rotation. Zerol teeth are curved but have zero spiral angle at mid-face, so they produce a straight bevel’s forces with a spiral’s smoothness. A HYPOID pair is something else and this page refuses it: its axes are offset rather than intersecting, its pitch surfaces are hyperboloids rather than cones, its pinion and wheel spiral angles are not equal, and the lengthwise sliding the offset introduces means it needs an API GL-5 extreme-pressure oil rather than a bevel gear oil. Nothing on this page applies to one.

The practical part, which is the part that gets skipped. A bevel pair is a matched set, usually lapped together and marked with a set number, and it is replaced as a set — a new pinion against an old wheel has a contact pattern nobody designed. It is not mounted to a centre distance; each member is shimmed to a stated mounting distance from its locating face to the common apex, both of which this page computes, and then the pinion is moved axially to set the backlash. The acceptance test is the contact pattern in marking compound: a patch centred on the tooth, clear of both ends. For the backlash itself see the contact ratio and backlash calculator, and for the shafts the pair’s thrust acts on, the shaft deflection and slope calculator.

Frequently asked questions

How do I calculate the pitch cone angle of a bevel gear?

For a right-angle pair, δ₁ = arctan(z₁/z₂) for the pinion and δ₂ = 90° − δ₁ for the wheel. For any other shaft angle Σ the general formula is δ₁ = arctan[sin Σ/(z₂/z₁ + cos Σ)], with δ₂ = Σ − δ₁. Both cases satisfy the rolling condition sin δ₂/sin δ₁ = z₂/z₁, which is worth checking because it catches a sign error immediately.

What is the maximum face width of a bevel gear?

The lesser of a cone-distance limit and a module limit, and the cone-distance one has two published forms: 0.30 Rₑ (MITCalc following ISO 23509, and Shigley’s 0.3 Aₒ) or Rₑ/3 (GlobalSpec’s reference). The module limit is 10 m, which Shigley writes as 10/P_d and GlobalSpec as three times the circular pitch — the same thing. A Zerol pair takes 25 per cent of the cone distance instead. Take the tighter figure if the two disagree, because the failure the rule prevents is a broken tooth at the small end.

Why do bevel gears fail at the small end?

Because the tooth there is smaller. A bevel tooth is a slice of a cone, so its module, height and thickness all scale with distance from the apex: at the recommended face width limit the small end is 70 per cent of the size at the large end, where the gear was dimensioned. The mesh loads the whole face anyway, so an over-wide face puts a large share of the load on a much weaker tooth. Mounting error makes it worse, because a slender small-end tooth is also less stiff and draws the contact toward itself.

Is a hypoid gear a type of bevel gear?

No, and treating it as one is a real error rather than a vocabulary one. A bevel pair’s axes intersect and its pitch surfaces are cones; a hypoid pair’s axes are OFFSET and never meet, so its pitch surfaces are hyperboloids. Its pinion and wheel do not have the same spiral angle, its pinion is usually the larger relative to the wheel than a bevel pinion would be, and the offset introduces lengthwise sliding along the tooth that a bevel mesh does not have — which is why a hypoid needs an API GL-5 extreme-pressure lubricant. None of the formulas on this page applies to one.

Can I replace just one gear of a bevel pair?

Not if you want it to last. Spiral and Zerol pairs are lapped together as a set and are normally marked with a matched-set number; the tooth surfaces have been run against each other to produce a specific contact pattern, and that pattern is what the pair’s load capacity depends on. A new member against an old one has whatever contact the accumulated errors of both happen to give, usually concentrated at one end of the tooth. Order both, and re-shim to the drawing’s mounting distances when you fit them.

How is backlash set on a bevel pair?

By shimming, and specifically by moving the PINION axially. The wheel is shimmed first to put its pitch cone apex on the axis intersection, which fixes the contact pattern’s position along the tooth; then the pinion is shimmed to set the backlash, because moving it along its own axis changes the backlash without disturbing the pattern much. Measure the backlash at the wheel with a dial indicator on a tooth flank, and check the contact pattern with marking compound afterwards — the two adjustments interact.

What is the mounting distance on a bevel gear drawing?

The axial distance from the member’s locating face — the face it seats against in the housing — to the apex of its pitch cone. Both members have one, and correct assembly means both apexes end up at the same point, which is the intersection of the two shaft axes. It is the bevel equivalent of a centre distance and it is the dimension that decides whether the contact pattern is where it should be. This page computes it as Rₑ·cos δ − hₐ·sin δ for each member.

Related calculators

References

ISO 23509:2016, Bevel and hypoid gear geometry. Cited by number. It is the document that defines the bevel geometry this page computes and, importantly, the one that treats bevel and hypoid geometry together while keeping them distinct — a hypoid pair has a hypoid offset, and its pinion and wheel spiral angles are not equal, which is why nothing on this page applies to one.
MITCalc, Bevel and hypoid gears according to ISO 23509 (module documentation). The source for one of the two published face-width rules used here, in its own words: “generally, the face width is 30 % of the cone distance R_e2 or 10 m_et, whichever is less”, together with the qualifications that a Zerol face width should be multiplied by 0.83 and not exceed 25 % of R_e2, and that a shaft angle above 90° wants a smaller face width than the rule gives.
GlobalSpec / CDS Mechanical Design — Bevel Gears reference page. The source for the OTHER published face-width rule: “the face width should not normally exceed either one third of the cone distance or three times the pitch”. Worth reading beside MITCalc’s, because the two cone-distance coefficients differ by 11 % and the two module coefficients — three circular pitches is 9.42 m against 10 m — are the same rule differently rounded.
Richard Budynas and Keith Nisbett, Shigley’s Mechanical Engineering Design, chapters 14 and 15. Cited for the four Barth velocity factors and for the bevel face-width rule b ≤ min(0.3 A_o, 10/P_d). The velocity factors were verified rather than transcribed: an independent implementation of them returns K_v = 2.311 for a cast profile at 4 m/s, which is (3.05 + 4)/3.05 to four figures, and the metric constants 3.05, 6.1, 3.56 and 5.56 are the exact unit conversions of the original 600 and 1200 ft/min and 50 and 78 √(ft/min).
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.
Kendall Motor Oil, Hypoid Gear Oil: What It Is and When to Use It. Cited for the practical consequence of the hypoid boundary: a hypoid mesh adds “a high degree of sliding motion on the gear tooth face” that a bevel mesh does not have, the film can be “squeezed out from between the meshing gear teeth”, and the lubricant is therefore an API GL-5 extreme-pressure oil rather than a bevel gear oil. It is the clearest reason why calling a hypoid a bevel gear is not a vocabulary quibble.
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.