Gear Ratio & Spur Gear Calculator
Tooth counts in — ratio, speed, torque, direction and gear dimensions out.
| Shaft | Gears | Speed | Torque | Turns |
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| Mesh | Ratio | Common factor | Same teeth meet |
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How it was calculated
Spur gear geometry
| Dimension | Pinion | Gear |
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| Pressure angle | Hobbed, no undercut | Two equal gears | For this ratio |
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About the Gear Ratio & Spur Gear Calculator
Enter the tooth counts of a gear train and the calculator gives the ratio — also as an exact fraction — the output speed and torque, and which way the output turns. It handles simple trains (gears meshing one after another, with idlers) and compound trains (pairs of gears on shared shafts), with a per-mesh efficiency and a speed and torque for every shaft.
The spur-gear section works out the reference (pitch), tip, root and base diameters, addendum, dedendum, circular pitch and centre distance from the module or diametral pitch, plus the contact ratio. It checks the pinion for undercut and the pair for interference at 14.5°, 20° and 25° pressure angles, and gives the profile shift that avoids undercut.
How to use it
- Choose a simple train (each gear meshes with the next) or a compound train (each stage’s driven gear shares a shaft with the next stage’s driver).
- Enter the tooth counts, from the input (driver) to the output. Add or remove gears or stages with the buttons.
- Optionally enter the input speed and torque and the efficiency of one mesh (98% is typical for spur gears) to get the output speed, torque and power.
- Under Spur gear geometry, pick module (mm) or diametral pitch (teeth per inch), the pressure angle and the tooth counts of the pinion and gear.
- Read the dimensions table and the undercut and interference checks. Copy result copies both sections as text.
Examples
20 → 30 → 60 teeth, 1,500 rpm, 10 N·m, 98% per mesh
i = 3 : 1, output 500 rpm, 10 × 3 × 0.98² = 28.81 N·m, same direction as the input
Stages 20/60 and 15/45, 1,500 rpm, 10 N·m
i = 3 × 3 = 9 : 1, output 166.7 rpm and 86.44 N·m; the middle shaft turns at 500 rpm the other way
m = 2 mm, 20° pressure angle, 20 and 40 teeth
Pitch diameters 40 and 80 mm, tip 44 and 84 mm, root 35 and 75 mm, centre distance 60 mm, contact ratio 1.64
m = 2 mm, 20°, 12 teeth
Undercut when hobbed (limit 17.1 teeth): profile shift x ≥ 0.298, or use 25° (limit 11.2 teeth)
The formulas
- Simple train: i = z_output ÷ z_input — the idlers in between cancel out, but each mesh reverses the direction
- Compound train: i = (z2 ÷ z1) × (z4 ÷ z3) × … — driven teeth over driver teeth, stage by stage
- Speed and torque: n_out = n_in ÷ i and T_out = T_in × i × η, where η = η_mesh^(number of meshes)
- Gear size: d = m × z, tip d_a = d + 2m, root d_f = d − 2.5m (full-depth teeth), base d_b = d × cos α
- Pitch: p = π × m; base pitch p_b = p × cos α; centre distance a = m × (z1 + z2) ÷ 2
- Diametral pitch: P = z ÷ d in inches, so m = 25.4 ÷ P (10 DP ≈ module 2.54)
- Contact ratio: ε_α = (√(r_a1² − r_b1²) + √(r_a2² − r_b2²) − a × sin α) ÷ (π × m × cos α)
Module, diametral pitch and pressure angle
The module is the pitch diameter per tooth in millimetres; ISO 54 lists preferred modules (series I: 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10 … 50) and second choices (series II: 1.125, 1.375, 1.75, 2.25 …). Inch gears use the diametral pitch, the number of teeth per inch of pitch diameter. Two gears mesh only if they have the same module (or DP) and pressure angle.
20° is today’s standard pressure angle (the ISO 53 basic rack). 25° gives stronger teeth and allows fewer teeth; 14½° is no longer recommended for new designs but is still made for replacements. Full-depth teeth have an addendum of 1 m and a dedendum of 1.25 m; stub teeth use 0.8 m and 1 m.
Undercut, interference and profile shift
When a gear with few teeth is cut by a hob or rack cutter, the cutter removes material at the root of the tooth (undercut), which weakens it and shortens contact. A standard gear avoids it with at least 2 ÷ sin²α teeth: 31.9 at 14½°, 17.1 at 20° and 11.2 at 25°. Shifting the cutter outward by x × m with x ≥ 1 − z × sin²α ÷ 2 avoids undercut on smaller gears (x = 0.298 for 12 teeth at 20°), but changes the tooth thickness and the centre distance.
Interference is the mating problem: the tip of one gear digs into the flank of a small pinion. Shigley gives the smallest pinion for each ratio — at 20°, 13 teeth for a 1 : 1 pair and 18 teeth to run with a rack; a 13-tooth pinion meshes without interference with gears up to 16 teeth, a 16-tooth pinion with up to 101.
Hunting teeth
If the two tooth counts share a common factor, the same teeth meet over and over: 20 and 60 teeth meet every 3 turns of the small gear. Counts with no common factor (such as 19 and 60) make every tooth meet every other tooth in turn, which evens out wear. The calculator shows this for every mesh.
Sources
- ISO 53:1998, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile
- ISO 54:1996, Cylindrical gears for general engineering and for heavy engineering — Modules
- ISO 21771, Gears — Cylindrical involute gears and gear pairs — Concepts and geometry
- ANSI/AGMA 1012-G05, Gear Nomenclature, Definitions of Terms with Symbols
- Budynas & Nisbett, Shigley’s Mechanical Engineering Design, 11th ed., ch. 13 — contact ratio (§13-6) and interference (§13-7)
Limitations
- Trains of external spur or helical gears on fixed shafts. Internal (ring) gears, planetary sets, worm gears and bevel gears are not covered.
- Geometry is for standard gears without profile shift; the tool gives the shift needed to avoid undercut but not the dimensions of shifted gears or backlash allowances.
- No strength rating: bending and contact stresses need ISO 6336 or ANSI/AGMA 2001 and the material data.
- The mesh efficiency is your input; real losses depend on load, speed, lubrication and accuracy.
Privacy
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Frequently asked questions
How do you calculate a gear ratio?
Divide the teeth on the driven gear by the teeth on the driver: a 20-tooth gear driving a 60-tooth gear gives 3 : 1, so the output turns three times slower and with about three times the torque. For a compound train, multiply the ratios of the stages.
Does an idler gear change the gear ratio?
No. In 20 → 30 → 60 the ratio is still 60 ÷ 20 = 3 : 1. The idler only reverses the direction (and sets the spacing), and its shaft carries no torque.
How do I convert diametral pitch to module?
m = 25.4 ÷ P. A 10 DP gear has a module of 2.54 mm and a 24 DP gear about 1.058 mm. Gears only mesh with the same module or DP and the same pressure angle, so a metric and an inch gear rarely mesh properly.
What is the minimum number of teeth on a spur gear?
For a standard full-depth gear cut by a hob: 18 teeth at 20° (the limit is 17.1, so 17 is very slightly undercut), 32 at 14½° and 12 at 25°. Fewer teeth need profile shift or a larger pressure angle.
How does a gear ratio change torque?
Output torque is the input torque times the ratio, less the losses: T_out = T_in × i × η. A 9 : 1 compound train with two 98% meshes turns 10 N·m into 10 × 9 × 0.96 = 86.4 N·m, while the speed drops to one ninth.
Which way does the output gear turn?
Each pair of external gears reverses the rotation. With an odd number of meshes the output turns the opposite way to the input; with an even number, the same way.