For a two-gear mesh, gear ratio is determined by comparing the number of teeth on the driving (input) gear to the number of teeth on the driven (output) gear, which indicates how much an output gear is sped up or slowed down or how much torque is lost or gained in the system [S2][S3].
The tooth-thickness and spacing terms cancel because meshing gears must share the same module (m) and pressure angle (α), typically 20° in modern ISO 6336 designs, so tooth geometry is forced to match and only the tooth count remains [S2][S5].
Why Teeth Replace Diameter in the Equation
The textbook ratio starts as circumference of output divided by circumference of input; because circumference = π × diameter, the π cancels, leaving diameter-output / diameter-input [S2]. Diametral pitch (DP, used in U.S. inch-unit design) and module m (used in ISO/AGMA mm-unit design) both fix a linear relationship between a gear's pitch diameter and its tooth count, so diameter and tooth count are interchangeable inputs to the same ratio [S5].
For an ISO gear with module m = 3 mm, the circular pitch p = πm = 9.4248 mm; transforming DP 8 to module gives m = 25.4 / 8 = 3.175 mm, showing how the two systems are numerically equivalent, not different gear geometries [S5]. The same equation also underlies the Spicer ring-and-pinion calculator used for axle swaps, where ring-gear teeth are divided by pinion teeth to express final drive numerically (for example 41 / 11 = 3.73) [S1].
Spur, Helical, Bevel, Planetary: Same Ratio, Different Geometry
Spur, helical, bevel, worm, and planetary gear sets all use the identical tooth-count ratio; the only thing that changes is shaft orientation, sliding losses, and the minimum tooth count needed to avoid undercut [S2]. Helical gears in a helical gear reducer use a normal module rather than transverse module, so the tooth count that appears in the ratio equation is the virtual (or transverse) tooth count at the operating pitch diameter, not the physical count on the blank.
For crossed-axis or worm pairs, the ratio is still driven-gear teeth / driver-tooth count, but a single-start worm acts as a "1-tooth" driver, so a 40-tooth worm wheel paired with a single-start worm is 40:1; doubling the worm starts halves the ratio for the same wheel [S2]. Planetary stages are an extension of the same idea: a planetary set with a 40-tooth ring gear, 20-tooth planet, and 20-tooth sun gives 3:1 in reduction (ring fixed) and the total ratio multiplies into the rest of the train when the planetary output is coupled to another shaft [S4].
Compound Trains: Multiply the Stages, Not the Teeth
In a compound train where the driven gear of stage 1 is on the same shaft as the driving gear of stage 2, total ratio = (Teeth_stage1_out / Teeth_stage1_in) × (Teeth_stage2_out / Teeth_stage2_in); the same-shaft gear does not contribute its own ratio, it merely transmits speed and torque across the shaft [S4]. A typical 2-stage helical industrial gear unit might run 20:1 by chaining a 3.16:1 first stage and a 6.33:1 second stage, both derived from integer tooth counts such as 60/19 and 95/15.
The Evolvent Design calculator convention is to tick a "same shaft" box on any gear that is locked to the one above it, leaving the box empty when the gear actually meshes, so the script skips the tooth-division step for compound couplings and only multiplies the real mesh ratios [S4]. Skipping this convention is the single most common reason a hand-built ratio spreadsheet disagrees with a vendor's published reduction number on a gear reducer nameplate.
Tooth Count Limits: Undercut, Pinion Minimum, Profile Shift
A 20° pressure-angle spur pinion needs at least about 17 teeth to avoid involute undercut, while a 14.5° pinion (older AGMA stock) needs roughly 32 teeth for the same condition, which is why modern ISO 6336 / AGMA 2015 stock gears default to 20° [S5]. Going below the minimum is possible with profile shift (x), but each 0.1 of positive shift moves the operating pressure angle and reduces the slip-free contact ratio, so tooth-count choice and shift coefficient must be solved together rather than independently.
Tooth depth itself is set by the module: full-depth ISO/JIS teeth use addendum ha = 1.00 m, dedendum hf = 1.25 m, total tooth depth h = 2.25 m, so a module-3 gear cuts 6.75 mm of flank and a module-5 gear cuts 11.25 mm [S5]. Higher module means stronger teeth but a larger pitch diameter for the same tooth count, which raises the gearbox envelope and the moment arm on the bearings, a trade-off that shows up in any gear coupling sized to the same shaft.
Reverse Calculation: Picking Teeth for a Target Ratio
To design for a target ratio, start by locking the smallest practical pinion (often 14 to 18 teeth for 20° spur), then solve Teeth_output = round(target × Teeth_input); accept the rounding and adjust the input count if the rounded output lands on a non-integer [S2][S5]. For a 5.667:1 target with an 18-tooth pinion, output = 5.667 × 18 = 102.0, which is achievable; with a 17-tooth pinion the same target asks for 96.3 teeth, and you would round to 96 and accept a true ratio of 5.647:1.
For a gear pump the same ratio logic applies in reverse: the displacement per revolution is fixed by tooth count and module, so a pump designer picks a tooth count that delivers the target cc/rev at the chosen center distance rather than picking a center distance and back-solving. Either way the meshing rule (same module, same pressure angle, teeth ratio = speed ratio) is what makes the gear set work at all, and it is the same rule Spicer uses to express an axle ratio from ring-and-pinion tooth counts [S1].
Failure Modes When the Math Is Wrong
The most common error is ignoring module matching: mating gears with the same tooth count but different modules will not roll without binding, no matter what the ratio "should" be [S2][S5]. The second is mixing diametral pitch and module on the same drawing: a DP 8 gear has module 3.175, a module 3 gear has DP 8.4667, and bolting them together produces a contact ratio below 1.0 and immediate tip-to-root interference [S5].
The third is treating a same-shaft compound stage as a reducer: the shaft-locked gear transmits torque at 1:1, and only the two real meshes contribute ratios; a spreadsheet that divides by every tooth pair in a 3-shaft train will read a ratio that is the square of the true value [S4]. A simple sanity check: the ratio should equal input speed / output speed measured at the shafts; if it does not, recheck the same-shaft markings before rechecking the teeth [S2].
Track the next revision of ISO 6336 for updated contact-ratio and bending-strength factors, and watch for AGMA 2015 errata on minimum pinion tooth counts for profile-shifted pairs; these documents govern the tooth-count range that any catalog spur, helical, or bevel set can ship within.
Related analysis: Pump-Drive Coupling Selection: Torque, DBSE, Misalignment Envelope.