This page collects the gear formulas for every common gear type in one reference: spur, helical, internal, rack and pinion, straight bevel, spiral bevel and worm. Module, diameters, pitch, centre distance, contact ratio and tooth forces are all here. Each section links to the calculator that performs that particular calculation for you — the table if you want the formula, the tool if you want the number.
All lengths are in millimetres and all angles in degrees. The tables give the standard (unshifted) case; how profile shift enters the calculation is covered in its own section at the end.
Symbols and units
| Symbol | Meaning | Unit |
|---|---|---|
| m | Module | mm |
| mₙ / mₜ | Normal module / transverse (radial) module | mm |
| z | Number of teeth | — |
| α | Pressure angle (20° standard) | ° |
| αₙ / αₜ | Normal / transverse pressure angle | ° |
| α_w | Working pressure angle | ° |
| β | Helix angle | ° |
| β_m | Spiral angle (spiral bevel gears) | ° |
| d | Pitch diameter | mm |
| d_b / d_w | Base diameter / working pitch diameter | mm |
| dₐ / d_f | Outside diameter / root diameter | mm |
| hₐ / h_f / h | Addendum / dedendum / whole depth | mm |
| p | Circular pitch | mm |
| a | Centre distance | mm |
| b | Face width | mm |
| x | Profile shift coefficient | — |
| y | Centre distance increment factor | — |
| i | Gear ratio | — |
| ε_α / ε_β / ε_γ | Transverse / overlap / total contact ratio | — |
| Σ | Shaft angle (bevel gears, usually 90°) | ° |
| δ | Pitch cone angle | ° |
| R_e | Cone distance | mm |
| γ | Lead angle of a worm | ° |
| inv α | Involute function: inv α = tan α − α (α in radians) | — |
Spur gear formulas
| Quantity | Symbol | Formula |
|---|---|---|
| Pitch diameter | d | z · m |
| Base diameter | d_b | d · cos α |
| Addendum | hₐ | 1.00 · m |
| Dedendum | h_f | 1.25 · m |
| Whole depth | h | 2.25 · m |
| Outside diameter | dₐ | d + 2m |
| Root diameter | d_f | d − 2.5m |
| Circular pitch | p | π · m |
| Centre distance | a | (z₁ + z₂) · m / 2 |
| Gear ratio | i | z₂ / z₁ |
| Module ⇄ DP conversion | DP | 25.4 / m |
Two gears mesh only if their modules and pressure angles match. To run these with your own inputs use the spur gear calculator, and to identify the module of a gear you already have, the gear module calculator.
Helical gear formulas
A helical gear lives in two planes: the normal plane perpendicular to the tooth, and the transverse plane of rotation. The table below follows the normal system, which is what most manufacturing uses, because one hob then serves every helix angle.
| Quantity | Symbol | Formula |
|---|---|---|
| Transverse module | mₜ | mₙ / cos β |
| Transverse pressure angle | αₜ | arctan(tan αₙ / cos β) |
| Pitch diameter | d | z · mₙ / cos β |
| Base diameter | d_b | d · cos αₜ |
| Outside / root diameter | dₐ / d_f | d + 2mₙ / dₐ − 4.5mₙ |
| Centre distance | a | (z₁ + z₂) · mₙ / (2 cos β) |
| Virtual number of teeth | z_v | z / cos³β |
| Normal / transverse pitch | pₙ / pₜ | π·mₙ / π·mₜ |
| Axial pitch | pₓ | pₙ / sin β |
| Lead | L | π · d / tan β |
| Normal ⇄ transverse system | — | mₜ = mₙ/cos β · αₜ = arctan(tan αₙ/cos β) · xₜ = xₙ·cos β |
On parallel shafts the helix angles must be equal and the hands opposite. The cutter number is chosen from the virtual tooth count z_v, not the actual one. Full explanation and calculation: helical gear calculator.
Internal gear formulas
An internal gear carries its teeth on the inside of the rim. The pinion (1) runs externally and the internal gear (2) internally, so the tip and root diameters of the internal member are defined in the opposite sense.
| Quantity | Symbol | Formula |
|---|---|---|
| Pitch diameter | d | z · m |
| Base diameter | d_b | d · cos α |
| Centre distance (unshifted) | a | (z₂ − z₁) · m / 2 |
| Whole depth | h | 2.25 · m |
| Pinion outside diameter | dₐ₁ | d₁ + 2hₐ₁ |
| Internal gear outside diameter | dₐ₂ | d₂ − 2hₐ₂ |
| Pinion root diameter | d_f₁ | dₐ₁ − 2h |
| Internal gear root diameter | d_f₂ | dₐ₂ + 2h |
| Working pitch diameter | d_w | d_b / cos α_w |
| Gear ratio | i | z₂ / z₁ (same direction of rotation) |
Three separate kinds of interference have to be checked on an internal mesh: involute interference, trochoid interference and trimming interference. The condition for avoiding involute interference is z₁/z₂ ≥ 1 − tan α_a2 / tan α_w, where α_a2 = arccos(d_b2 / dₐ2) is the pressure angle at the tip of the internal tooth; this holds only when dₐ2 ≥ d_b2. The working pressure angle follows from α_w = arccos[(z₂ − z₁)·m·cos α / (2a)].
Note that internal gears are usually cut with a pinion-type cutter. Since the cutter itself carries a profile shift coefficient, the actual tooth depth and root diameter after cutting differ slightly from the calculated values; for precision work the cutter’s coefficient has to be taken into account.
Rack and pinion formulas
A rack can be thought of as a gear of infinite diameter: the pitch circle becomes a straight line and the tooth flanks become straight. The pinion follows the spur gear table exactly; on the rack side the following apply.
| Quantity | Symbol | Formula |
|---|---|---|
| Rack pitch | p | π · m |
| Rack addendum | hₐ | 1.00 · m |
| Rack whole depth | h | 2.25 · m |
| Pinion pitch diameter | d | z · m |
| Centre distance (pinion axis to rack base) | a | z·m / 2 + H (H: pitch line height) |
| Linear travel per pinion revolution | — | π · d = π · m · z |
| Pinion revolutions for a given travel | — | travel / (π · m · z) |
The output most often wanted from a rack and pinion drive is the answer to “how far does the axis move in one pinion revolution”: π·m·z. A module 2, 20 tooth pinion travels 125.66 mm per turn. On a helical rack the pitch is measured in the normal plane and the same relations hold with mₙ. Run the numbers with our rack and pinion calculator.
Straight bevel gear formulas
Two systems are in common use for bevel gears and they give different results from the same inputs. That is not a contradiction but two separate design traditions; decide which system the gear will be made to before you start, and never mix the tables.
| Quantity | Symbol | Gleason system | Standard system |
|---|---|---|---|
| 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) | |
| Face width limit | b | b ≤ R_e/3 and b ≤ 10m | |
| Gear addendum | hₐ₂ | 0.540m + 0.460m / [(z₂cos δ₁)/(z₁cos δ₂)] | 1.00 m |
| Pinion addendum | hₐ₁ | 2.000m − hₐ₂ | 1.00 m |
| Dedendum | h_f | 2.188m − hₐ | 1.25 m |
| Dedendum angle | θ_f | arctan(h_f / R_e) | |
| Addendum angle | θₐ | θₐ₁ = θ_f₂ , θₐ₂ = θ_f₁ | arctan(hₐ / 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 | |
The difference between the two systems lives entirely in the tooth heights. The Gleason system deliberately splits the addendum unevenly — longer on the pinion, shorter on the gear — so the weaker small member is strengthened. In the standard system both are 1.00m. The Gleason system also sets each member’s addendum angle equal to the other’s dedendum angle, which keeps the clearance constant along the cone. The bevel gear calculator covers all three systems.
Spiral bevel gear formulas
A spiral bevel gear is the curved-tooth version of a straight bevel gear, giving the bevel mesh what a helix gives a spur mesh. The calculation sequence is the same as the Gleason straight bevel, with the pressure angle corrected for the spiral angle and different tooth height coefficients.
| Quantity | Symbol | Formula |
|---|---|---|
| Transverse pressure angle | αₜ | arctan(tan αₙ / cos β_m) |
| Pitch diameter | d | z · m (m: outer transverse module) |
| Cone angles | δ₁, δ₂ | arctan[ sin Σ / (z₂/z₁ + cos Σ) ] , Σ − δ₁ |
| Cone distance | R_e | d₂ / (2 sin δ₂) |
| Gear addendum | hₐ₂ | 0.460m + 0.390m / [(z₂cos δ₁)/(z₁cos δ₂)] |
| Pinion addendum | hₐ₁ | 1.700m − hₐ₂ |
| Dedendum | h_f | 1.888m − hₐ |
| Remaining quantities | θ, δ, dₐ, X, X_b, d_i | same as the Gleason straight bevel table |
The standard spiral angle is 35°, and a working pair has opposite spiral hands. With the spiral angle set to zero the gear becomes a Zerol type and the straight bevel table applies. Hypoid gears fall outside these tables entirely: their axes do not intersect but are offset, and the established engineering standards do not cover hypoid geometry — which is why no hypoid formulas are given here.
Worm and worm wheel formulas
| Quantity | Symbol | Formula |
|---|---|---|
| Worm pitch diameter | d₁ | Q · mₓ (Q: diameter factor = d₁/mₓ) |
| Wheel pitch diameter | d₂ | z₂ · mₓ |
| Lead angle | γ | arctan(mₓ · z_w / d₁) |
| Centre distance | a | (d₁ + d₂) / 2 |
| Addendum | hₐ | 1.00 · mₓ |
| Whole depth | h | 2.25 · mₓ |
| Worm outside diameter | dₐ₁ | d₁ + 2hₐ₁ |
| Wheel throat diameter | d_th | d₂ + 2hₐ₂ |
| Wheel outside diameter | dₐ₂ | d₂ + 2hₐ₂ + mₓ |
| Root diameters | d_f₁ / d_f₂ | dₐ₁ − 2h / d_th − 2h |
| Gear ratio | i | z₂ / z_w |
Here mₓ is the axial module and z_w the number of worm threads (starts). A worm drive’s ratio is set by the thread count, not by a tooth count on the worm: a single-start worm with a 30 tooth wheel gives 30:1. A sufficient threaded length for the worm is b₁ = π·mₓ·(4.5 + 0.02·z₂), and the working blank width of the wheel is b_e = 2mₓ·√(Q + 1), so the actual blank should be b ≥ b_e + 1.5mₓ. Note that several methods exist for the wheel outside diameter; the table gives the common one.
Contact ratio formulas
The contact ratio is the length of the line of action divided by the base pitch — in practice, the answer to “how many tooth pairs carry load at any instant, on average”. Good practice keeps it at 1.2 or above, and never below 1.1.
| Mesh type | Formula |
|---|---|
| Spur pair (ε_α) | [√((dₐ₁/2)² − (d_b₁/2)²) + √((dₐ₂/2)² − (d_b₂/2)²) − a·sin α_w] / (π·m·cos α) |
| Helical pair, transverse part (ε_α) | Same expression, with π·mₜ·cos αₜ in the denominator and a·sin α_wt |
| Overlap (ε_β) | b · sin β / (π · mₙ) |
| Total (ε_γ) | ε_α + ε_β |
| External–internal pair | Spur expression with the second root subtracted and a·sin α_w added |
There are three ways to raise a contact ratio: decrease the pressure angle, increase the number of teeth, or increase the working tooth depth. The overlap component exists only in helical and spiral tooth forms; on a spur gear ε_β is zero.
Gear force formulas
| Gear type | Tangential force F_u | Axial force F_a | Radial force F_r |
|---|---|---|---|
| Spur gear | 2000 · T / d | — | F_u · tan α |
| Helical gear | 2000 · T / d | F_u · tan β | F_u · tan αₙ / cos β |
| Straight bevel gear | 2000 · T / d_m | F_u · tan α · sin δ | F_u · tan α · cos δ |
| Spiral bevel, convex flank working | 2000 · T / d_m | (F_u/cos β_m)·(tan αₙ sin δ − sin β_m cos δ) | (F_u/cos β_m)·(tan αₙ cos δ + sin β_m sin δ) |
| Spiral bevel, concave flank working | 2000 · T / d_m | (F_u/cos β_m)·(tan αₙ sin δ + sin β_m cos δ) | (F_u/cos β_m)·(tan αₙ cos δ − sin β_m sin δ) |
Units: torque T in N·m, diameter d in mm, force in N. On bevel gears the calculation uses the central pitch diameter, d_m = d − b · sin δ. Pitch line velocity is v = π · d · n / 60000 [m/s], with n in rpm.
On a helical gear the axial force scales directly with the helix angle: 0.27 times the tangential force at 15°, 0.58 times at 30°, and all of it at 45°. That force goes into the bearings and drives the bearing type selection.
Undercut and profile shift
Below a certain tooth count the cutter digs into the root of the tooth and thins its waist — this is undercut. It weakens the tooth and removes useful involute next to the base circle. For an unshifted spur gear the limiting condition is:
| Quantity | Formula |
|---|---|
| No-undercut condition | hₐ ≤ (d/2) · sin²α |
| Minimum tooth count without undercut | z_c ≥ 2 / sin²α → 18 at α = 20°, 32 at α = 14.5° |
| Limit with profile shift | z_c = 2(1 − x) / sin²α |
| Shift required for a given z | x = 1 − (z_c/2) · sin²α |
Profile shift is not only an anti-undercut measure; it is also used to force the centre distance onto a required value. In a shifted pair the sequence is: inv α_w = 2 tan α · (x₁ + x₂)/(z₁ + z₂) + inv α gives the working pressure angle, then the centre distance increment factor y = [(z₁+z₂)/2] · (cos α / cos α_w − 1) and the centre distance a = [(z₁+z₂)/2 + y] · m. A positive shift thins the tooth tip, so the top land thickness needs checking too.
On helical gears the undercut limit drops below the spur value, because the transverse pressure angle is larger than the normal one. That is why helical gears with very low tooth counts are practical.
Frequently asked questions
Which formula do you start a gear calculation from?
Module and tooth count are the two primitives; everything else follows from them. If you have both, start straight at d = z·m. If instead you have a centre distance and a speed ratio, you first have to derive the tooth counts — and those will not come out as whole numbers, so you round to the nearest integers and close the resulting gap with profile shift.
What is the difference between module and diametral pitch?
Both express tooth size, but they run in opposite directions. Module is the pitch diameter per tooth in the metric system, so a bigger module means a bigger tooth. Diametral pitch is the number of teeth per inch of pitch diameter, so a bigger DP means a smaller tooth. The conversion is DP = 25.4 / m.
Why are there two different formula sets for bevel gears?
The Gleason and standard systems are two separate design traditions. Gleason splits the addendum unevenly — long on the pinion, short on the gear — to strengthen the weaker small member. The standard system gives both 1.00m. Getting different results from the same inputs is expected; what matters is picking one system and carrying the whole calculation through with it.
Why are no hypoid formulas given?
A hypoid pair has offset, non-intersecting axes. That geometry falls outside the standard calculation methods developed for bevel gears, and a correct solution needs dedicated loaded tooth contact analysis software. Our sources do not contain a complete formula set for hypoid geometry, so we do not publish a guessed table here.
Which tooth profile do these formulas assume?
The tables are for the standard full-depth involute profile: addendum 1.00m, whole depth 2.25m. Stub, long-addendum or special profiles use different height coefficients, and there you have to substitute your own profile data.
The formulas on this page are intended for education and preliminary design; verify the results before manufacturing. Tools that run the numbers for you: spur gear calculator, helical gear calculator, rack and pinion calculator, bevel gear calculator and gear module finder. Cutting speeds and feeds for the machining itself are in our CNC speed and feed calculator.