A bevel gear calculator solves the cone geometry of a pair that connects intersecting shafts. The tool below takes module and tooth counts and returns the pitch cone angles, cone distance, addenda and dedenda, outer and root cone angles, outside diameter and the pitch-apex-to-crown dimension needed for assembly. Three systems are supported separately: Gleason straight bevel, standard straight bevel and Gleason spiral bevel. The same inputs give different results in each — not an error, but three distinct design traditions.
What is a bevel gear used for?
A bevel gear carries its teeth on a cone surface and connects intersecting shafts. Where spur and helical gears link parallel shafts, a bevel pair turns the drive through an angle — most often 90°. Differentials, right-angle gearboxes, drill and milling heads and hand-tool gearboxes all work on this principle.
The underlying geometry is this: two cones share an apex and roll on each other without slipping. As a result every dimension on a bevel gear changes along the cone — larger at the outer end, smaller at the inner. The calculations are referenced to the outer (heel) section, and the module is the module of that section.
How the cone angles are calculated
The first step of any bevel calculation is the pitch cone angles. Given the tooth counts and the shaft angle Σ:
δ₁ = arctan[ sin Σ / (z₂/z₁ + cos Σ) ] and δ₂ = Σ − δ₁
At a 90° shaft angle this simplifies to δ₁ = arctan(z₁/z₂). For a 20 and 40 tooth pair the pinion cone angle is arctan(0.5) = 26.565° and the gear's is 63.435°. The cone distance then follows from the outer pitch diameter: R_e = d₂ / (2 sin δ₂). This is the slant distance from the cone apex to the outer end, and it is the quantity the face width limits are built on.
The calculator handles shaft angles other than 90° as well, since the formula covers them. One caveat is worth knowing: the strength tables for bevel gears are given only for Σ = 90°. Choose a different angle and the geometry still comes out right, but the strength side needs separate treatment.
Gleason system or standard system?
Two design traditions are in common use for bevel gears, and the difference between them lives entirely in the tooth heights.
In the standard system both members get an addendum of 1.00m and a dedendum of 1.25m, exactly as on a spur gear. Simple and symmetric.
In the Gleason system the addendum is deliberately split unevenly: long on the pinion, short on the gear. The engineering reason is direct — the small member always makes more cycles and has the weaker tooth, and lengthening its addendum strengthens it. The coefficients are:
| Quantity | Gleason (straight) | Standard | Gleason (spiral) |
|---|---|---|---|
| Gear addendum hₐ₂ | 0.540m + 0.460m / K | 1.00 m | 0.460m + 0.390m / K |
| Pinion addendum hₐ₁ | 2.000m − hₐ₂ | 1.00 m | 1.700m − hₐ₂ |
| Dedendum h_f | 2.188m − hₐ | 1.25 m | 1.888m − hₐ |
Here K = (z₂ · cos δ₁) / (z₁ · cos δ₂). The second distinguishing rule of the Gleason system is that each member's addendum angle is set equal to the other's dedendum angle (θₐ₁ = θ_f₂ and θₐ₂ = θ_f₁), which keeps the root clearance constant along the cone. In the standard system each gear takes its own angle from its own addendum.
Which system you use is decided by your manufacturing route. What matters is carrying the whole calculation through with one system and never mixing values between them. The three systems are separate options in the calculator and are never combined in one output.
Bevel gear formulas
| Quantity | Symbol | Formula |
|---|---|---|
| Pitch diameter | d | z · m |
| Pinion cone angle | δ₁ | arctan[ sin Σ / (z₂/z₁ + cos Σ) ] |
| Gear cone angle | δ₂ | Σ − δ₁ |
| Cone distance | R_e | d₂ / (2 sin δ₂) |
| Dedendum angle | θ_f | arctan(h_f / R_e) |
| Outer cone angle | δₐ | δ + θₐ |
| Root cone angle | δ_f | δ − θ_f |
| Outside diameter | dₐ | d + 2hₐ · cos δ |
| Pitch apex to crown | X | R_e cos δ − hₐ sin δ |
| Axial face width | X_b | b · cos δₐ / cos θₐ |
| Inner outside diameter | d_i | dₐ − 2b sin δₐ / cos θₐ |
| Transverse pressure angle (spiral) | αₜ | arctan(tan αₙ / cos β_m) |
Symbols: m outer module, z tooth count, Σ shaft angle, δ pitch cone angle, δₐ outer cone angle, δ_f root cone angle, θₐ addendum angle, θ_f dedendum angle, R_e cone distance, b face width, hₐ addendum, h_f dedendum, X pitch apex to crown, β_m spiral angle. Lengths in millimetres, angles in degrees.
Of these, the pitch apex to crown dimension X is the one most often overlooked on the shop floor and the most critical at assembly: it is the axial distance from the cone apex to the front face, and it fixes where the gear sits on its shaft. If the two apexes do not meet at a single point, the contact pattern shifts, the pair gets noisy and the teeth wear on one edge.
Spiral bevel gears
On a spiral bevel gear the teeth follow a curved path across the cone rather than a straight one. It gives the bevel mesh what a helix gives a spur mesh: contact builds gradually, more teeth share the load, the drive is quieter and it carries more. The standard spiral angle is 35°, and a working pair has opposite spiral hands.
The calculation sequence matches the Gleason straight bevel with two differences. First, the pressure angle is corrected for the spiral angle: αₜ = arctan(tan αₙ / cos β_m). Second, the tooth height coefficients change (last column of the table above). With the spiral angle reduced to zero the gear becomes a Zerol type and the straight bevel table applies.
Face width and undercut limits
Face width cannot be chosen freely on a bevel gear. Because the cone narrows inward, a wider face makes the inner end of the tooth so small that it becomes difficult to cut. Two limits apply and both must be satisfied:
b ≤ R_e / 3 and b ≤ 10 · m
The calculator checks your face width against both and warns you when either is exceeded.
Undercut on bevel gears is not governed by a single limiting number as on spur gears, but by a table of tooth count pairs. For straight bevel gears at a 20° pressure angle the accepted combinations are: a 16 tooth pinion needs at least 16 on the mate; 15 needs at least 17; 14 needs at least 20; 13 needs at least 30. Spiral bevels at β_m = 35° allow lower counts still: 17/17, 16/18, 15/19, 14/20, 13/22 and 12/26. The calculator checks your pair against these tables and warns you when the combination is not supported.
Worked example, step by step
Gleason straight bevel system, m = 3 mm, α = 20°, Σ = 90°, z₁ = 20, z₂ = 40, b = 22 mm.
- Pitch diameters: d₁ = 60 mm, d₂ = 120 mm
- Cone angles: δ₁ = arctan(1/2) = 26.565° · δ₂ = 63.435°
- Cone distance: R_e = 120 / (2·sin 63.435°) = 67.082 mm
- Face width check: R_e/3 = 22.36 mm and 10m = 30 mm → b = 22 mm is under both limits ✔
- Addenda: K = (40·cos 26.565°)/(20·cos 63.435°) = 4.0 → hₐ₂ = 0.540·3 + 0.460·3/4 = 1.965 mm · hₐ₁ = 6 − 1.965 = 4.035 mm
- Dedenda: h_f₁ = 2.188·3 − 4.035 = 2.529 mm · h_f₂ = 4.599 mm
- Angles: θ_f₁ = 2.159° · θ_f₂ = 3.922° → θₐ₁ = 3.922° · θₐ₂ = 2.159° (the Gleason cross rule)
- Outer and root cone angles: δₐ₁ = 30.487° · δ_f₁ = 24.406°
- Outside diameters: dₐ₁ = 67.218 mm · dₐ₂ = 121.758 mm
- Pitch apex to crown: X₁ = 58.196 mm · X₂ = 28.243 mm
Run the same inputs through the standard system and the addendum becomes 3.000 mm on both members, moving the outside diameters to 65.367 and 122.683 mm. The difference looks small on paper but produces a different part; components made to the two systems are not interchangeable.
Why hypoid gears are outside this tool
A hypoid gear looks like a spiral bevel at first glance but differs in one fundamental way: its axes do not intersect, they are offset. The crown-wheel-and-pinion set in an automotive differential is usually hypoid, and the offset is what lets the propeller shaft pass below the axle centreline.
That offset takes the geometry outside the scope of the bevel gear formulas. The established engineering standards developed for bevel gears do not cover hypoid geometry, and a correct solution requires software that simulates tooth contact under load. There is therefore no hypoid option in this tool: our sources do not contain a complete formula set for it, and we do not publish a guessed calculation.
Frequently asked questions
Where is the module of a bevel gear measured?
At the outer end, the wide side of the cone. Since every dimension varies along the cone, the section the module refers to matters; standard practice is the outer (heel) section, and that is what this tool uses. The relation d = z·m applies at that section.
What has to match for two bevel gears to run together?
Equal modules and pressure angles, cone angles that sum to the shaft angle, and both cone apexes meeting at a single point. Spiral bevels additionally need equal spiral angles with opposite hands. At assembly, the pitch apex to crown dimension X is what makes that meeting happen.
Why do the same inputs give two different answers?
Because Gleason and standard are two separate design traditions, not two versions of one calculation. Gleason splits the addendum unevenly to strengthen the pinion; the standard system splits it equally. There is no contradiction — you decide up front which system the gear will be made to and carry the whole calculation through with it.
Straight bevel or spiral bevel?
Straight bevels are simpler and cheaper to make and run fine at low speeds. Spiral bevels are quieter, carry more load for the same size and are clearly better at high speed, but they need dedicated machinery and they generate thrust. As a rule, the higher the pitch line velocity, the stronger the case for spiral.
Why can't I keep increasing the face width?
Because the cone narrows inward, so a wider face means a smaller tooth at the inner end, and past a point both cutting and strength suffer. That is why the limits b ≤ R_e/3 and b ≤ 10m exist. If you need a wider face, increase the module or the tooth counts so the cone distance grows with it.
The tool and formulas on this page are intended for education and preliminary design; verify the results before manufacturing. Related tools: spur gear calculator, helical gear calculator, rack and pinion calculator, and the combined gear formulas reference.