Spur gears have straight teeth parallel to their shaft axes. A meshing external pair transfers rotation between parallel shafts in opposite directions, with speed ratio set by the tooth counts. Change the two counts below to compare output speed and ideal torque multiplication and watch the gears turn.
Spur Gears Interactive Calculator
Vary the pinion and gear tooth counts to see the speed ratio, ideal torque multiplication, and animated gear mesh.
Equation Used
The spur gear ratio is set by tooth count: the driven gear teeth divided by the pinion teeth. A 20-tooth gear driven by a 12-tooth pinion gives i = 20 / 12 = 1.67, so the driven shaft turns at about 60% of input speed while ideal torque is multiplied by 1.67.
- The teeth maintain the angular ratio; the contacting flanks generally both roll and slide, with zero sliding at the pitch point.
- Z1 identifies the input gear and Z2 the output gear. The smaller member is commonly called the pinion; either member can drive.
- Ideal torque multiplier ignores bearing, windage, and tooth friction losses.
- The shortfall card compares Z1 with a 17-tooth reference for standard 20° gearing. It is a screening indicator, not a complete interference or manufacturing check; cutter geometry and profile shift also matter.
How a spur-gear pair works
The input tooth pushes against the output tooth along their common normal, called the line of action. For involute flanks this line is tangent to both base circles. Its geometry gives a constant angular-velocity ratio while contact moves along the tooth faces.
Tooth contact combines rolling and sliding. At the pitch point the sliding speed is zero; elsewhere the flanks slide relative to each other. Lubrication, surface finish, load and material therefore affect losses and wear even though the tooth-count ratio is fixed.
Parts and dimensions
- Input and output gears: exchange tangential force through their teeth. The smaller member is usually called the pinion.
- Module: the reference pitch diameter divided by tooth count, expressed in millimeters. A mating standard pair uses compatible module and pressure angle.
- Reference pitch circles: define nominal tooth spacing and, for an unshifted standard pair, the reference center distance.
- Pressure angle: sets the direction of tooth force relative to the pitch-circle tangent. The illustrative animation uses 20°.
- Backlash: clearance between mating flanks. Its required value depends on gear dimensions, tolerances, temperature and application.
- Shafts, bearings and housing: maintain alignment and react tooth forces.
Contact ratio describes the average number of tooth pairs in contact. Assess it with the actual profiles and operating center distance. This calculator does not determine contact ratio, tooth stress, bearing loads or service life.
Where spur gears are useful
A fixed-axis pair is useful when adjacent parallel shafts need a definite rotational ratio. Several pairs can form a compound reduction, while an idler can alter direction or bridge shaft spacing. Spur teeth also appear in planetary trains, where moving gear axes require a different speed relation from the fixed-axis pair shown here.
Choose a physical gear set using its rated loads, geometry, material, mounting and lubrication requirements. Noise, efficiency and usable speed depend on the particular design and duty; straight teeth alone do not establish those ratings.
Ratio, speed, torque and center distance
For input tooth count z₁ and output tooth count z₂, the reduction ratio is i = z₂/z₁. The output-speed magnitude is n₂ = n₁/i. External gears turn in opposite directions, so a signed angular-velocity equation is ω₂ = −ω₁z₁/z₂.
Ignoring losses, power balance gives output torque magnitude T₂ = iT₁. With efficiency η for this stage, T₂ = ηiT₁. Efficiency is not an input to this calculator, so its torque card shows the ideal multiplier only.
For standard unshifted metric gears, reference diameter d = mz and reference center distance a = m(z₁+z₂)/2, where m is module in millimeters. For imperial diametral pitch P in teeth per inch, d = z/P in inches. The conversion is P = 25.4/m when m is in millimeters.
Operating center distance and working pressure angle may differ for profile-shifted gears. Use the complete gear specification rather than treating the reference formula as a housing tolerance. Pitch-line speed is v = πdn/60, with d in meters and n in rpm.
Worked example: a 5:1 reduction
Consider a 21-tooth input gear driving a 105-tooth output gear. The ratio is 105/21 = 5. At an illustrative input speed of 550 rpm, output speed is 110 rpm in the opposite direction. The ideal torque multiplier is five; actual torque is lower according to losses.
For module 4 mm and standard unshifted geometry, the reference pitch diameters are 4×21 = 84 mm and 4×105 = 420 mm. The reference center distance is (84+420)/2 = 252 mm.
Pitch-line speed is π×0.084×550/60 ≈ 2.42 m/s. Using the output gear gives the same result: π×0.420×110/60 ≈ 2.42 m/s. Use each gear’s own diameter with its own speed.
The tooth-count calculator provides ratio, relative speed and ideal torque multiplication. Module and absolute input speed in this example are additional assumptions used for the hand calculation; they do not establish load capacity, lubricant choice, noise or life.
Comparing transmission arrangements
| Arrangement | Useful distinction | Design considerations |
|---|---|---|
| Spur gears | Straight teeth transfer rotation between parallel axes. | Ideal tooth force has tangential and radial components but no axial component. Backlash and tooth-contact behavior matter. |
| Parallel-axis helical gears | Inclined teeth engage progressively across the face. | Helix angle introduces axial thrust for a single helical pair. Bearing support and lubrication must suit the design. |
| Synchronous belt | A toothed belt connects separated pulleys without direct gear contact. | Belt tension, tooth engagement, compliance and pulley alignment affect load and positioning. |
Compare actual products for efficiency, accuracy, cost and life. None of these categories has a universal speed limit or a universally best reduction ratio.
Spur-gear questions
Tooth errors, changing mesh stiffness, alignment, backlash, structural vibration and load can all contribute. Noise alone does not diagnose low contact ratio or identify a repair. Inspect the complete drive and compare it with its specifications.
Compare required ratio, tooth counts, gear diameters, shaft spacing, loads and packaging. Two stage ratios multiply, but the extra mesh and bearings add losses and components. Ratio alone does not determine undercut or tooth strength.
Contact stress, material condition, surface finish, alignment, contamination and lubrication can affect pitting. Check the actual load history and manufacturing specifications before selecting a corrective action. This tooth-count model does not predict fatigue life.
It is a geometric starting point for the specified gear pair. Manufacturing tolerances, profile shifts, backlash, shaft deflection and thermal behavior must be considered before specifying the assembled center distance.
Do not assume they are compatible. Matching standard spur gears requires compatible tooth pitch and pressure-angle geometry as well as suitable profiles and clearance. Verify the manufacturer specifications; appearance is not a reliable identification method.
No. A preload can take up clearance only within the arrangement’s load range. Tooth and shaft deflection, bearing movement and assembly errors can still affect position. Increasing preload also changes contact and bearing loads; use the design’s rating rather than simply fitting a stiffer spring.
References and further reading
KHK: Calculation of Gear Dimensions covers standard and profile-shifted gear geometry. The KHK gear catalog provides product-specific dimensions, materials, accuracy and strength information.
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