Gear Basics and Terminology
Module and pitch, pressure angle, tooth terms, speed and torque ratios, contact ratio, undercut and how two gears mesh.
What gears do
A gear is a toothed wheel that transmits motion and torque by meshing with another toothed part. Gears give a fixed speed ratio with no slip (unlike a friction drive or a belt), change the speed and torque between shafts, change the direction of rotation, and can turn rotation into linear motion (a rack and pinion). They are compact and efficient, which is why they are found in everything from watches to wind turbines. This guide goes from the terms and geometry in this page, through the types, the strength calculation and gear trains, to the neighboring parts of a power-transmission system: belts, chains, shafts, bearings and lubrication. The motors that drive them are covered in Motion Control and Actuators.
Tooth terms and the module
| Term | Symbol and formula | Meaning |
|---|---|---|
| Module | m = d ÷ z (mm) | Pitch diameter per tooth: the size of the tooth. The metric standard |
| Diametral pitch | P = z ÷ d (teeth per inch) = 25.4 ÷ m | The inch equivalent; a larger P is a finer tooth |
| Circular pitch | p = π m | Distance between neighboring teeth along the pitch circle |
| Pitch diameter | d = m z | The diameter of the imaginary rolling circle where the two gears touch |
| Addendum, dedendum | ha = m; hf = 1.25 m | Tooth height above and below the pitch circle (standard full depth: whole depth 2.25 m) |
| Outside, root diameter | da = m (z + 2); df = m (z − 2.5) | The tip and root circles |
| Pressure angle | φ = 20° (14.5° older) | The angle between the line of action and the common tangent; sets the tooth shape |
| Base circle | db = d cos φ | The circle from which the involute is generated |
| Face width | b | The length of the tooth along the axis; sets the strength (typically 8 to 12 m) |
| Center distance | a = m (z1 + z2) ÷ 2 | Distance between the two shafts for a standard pair |
The involute is the curve traced by the end of a taut string unwinding from the base circle. Teeth shaped as involutes keep a constant speed ratio even if the center distance varies a little, and the common tooth force always acts along one fixed line (the line of action, see the forces in Gear Strength and Design Calculations).
How two gears mesh
Two gears mesh correctly only if they share the same module (or diametral pitch) and the same pressure angle. The pitch circles roll on each other without slipping, and the point where they touch is the pitch point. The smaller gear of a pair is the pinion, the larger the gear (or wheel). External gears turn in opposite directions; an internal (ring) gear turns the same way as its pinion.
Worked example: a 2 mm module pair
- Pinion z1 = 20 and gear z2 = 50, module m = 2 mm, φ = 20°.
- Pitch diameters: d1 = 2 × 20 = 40 mm, d2 = 100 mm. Center distance a = (40 + 100) ÷ 2 = 70 mm.
- Circular pitch p = π × 2 = 6.28 mm. Outside diameters da1 = 44 mm and da2 = 104 mm; root diameters df1 = 35 mm and df2 = 95 mm; base diameters 37.59 mm and 93.97 mm.
- In diametral-pitch terms the module of 2 mm is P = 25.4 ÷ 2 = 12.7 teeth per inch (close to DP 12.7: gears are not interchangeable between 2 mm module and DP 12 exactly).
Speed, torque and ratio
i = z2 ÷ z1 = d2 ÷ d1 = n1 ÷ n2 T2 = T1 × i × η P = T × ω
The gear ratio i is the number of teeth on the driven gear divided by the number on the driver. A ratio above 1 reduces the speed and multiplies the torque, a ratio below 1 does the opposite (an overdrive). Power stays the same apart from the losses in the mesh (efficiency η, typically 96 to 99 percent for one spur-gear mesh): speed down, torque up.
Worked example: speed and torque through the pair
- The ratio of the pair above is i = 50 ÷ 20 = 2.5.
- With the pinion at 1 500 rpm and 5 N·m (power 785 W), the gear turns at 1 500 ÷ 2.5 = 600 rpm.
- The ideal torque is 5 × 2.5 = 12.5 N·m; with 97 percent efficiency it is 12.1 N·m. The output power 762 W is 97 percent of the input: the other 24 W is lost as heat in the mesh and bearings.
- The Gear Ratio, Torque and Speed Generator does this arithmetic for a single pair or a train.
Contact ratio
While one pair of teeth is leaving contact the next pair must already have started, or the drive would run in jerks. The contact ratio ε is the average number of tooth pairs in contact: ε = (√(ra1² − rb1²) + √(ra2² − rb2²) − a sin φ) ÷ (π m cos φ). Standard spur gears have ε between 1.2 and 2; a value above 1.4 is wanted for quiet running, and 1 is the absolute minimum.
For the pair above: ε = (11.44 + 22.28 − 23.94) ÷ 5.90 = 1.66: between one and two pairs of teeth share the load at any moment, which is good. Helical gears (see Gear Types and Selection) add an axial overlap that raises the effective ratio.
Undercut and the minimum number of teeth
If a pinion has too few teeth the cutter removes the root of the tooth (undercutting), which weakens it, and the tip of the mating tooth can touch the root outside the involute (interference). The minimum number of teeth for a standard full-depth 20° tooth without undercutting is zmin = 2 ÷ sin²φ = 2 ÷ sin²20° = 17.1, so 17 teeth. In practice pinions of 14 to 16 teeth are used with a small amount of undercut, and smaller ones (down to 8 to 10, as in a gear head) are made with a profile shift (positive addendum modification) that moves the cutter outwards to avoid it and strengthen the tooth.
- Fewer teeth give a smaller drive but weaker, noisier teeth and a lower contact ratio.
- More teeth give smoother action, a higher contact ratio and a stronger tooth for the same module, at a larger diameter.
- Rack: a rack is a gear of infinite diameter; its teeth have straight flanks, and zmin does not apply (see the rack figure above).
Standard modules and pitches
| Series | Values |
|---|---|
| Metric module (ISO 54), preferred | 0.5, 0.8, 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20 mm |
| Metric module, second choice | 0.6, 0.7, 0.9, 1.125, 1.375, 1.75, 2.25, 2.75, 3.5, 4.5, 5.5, 7, 9, 11, 14, 18 mm |
| Diametral pitch (inch) | 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 48, 64 teeth per inch |
| Small gears for makers | Module 0.5 and 1 (about DP 50 and 25); 32 DP and 48 DP in model work |
Use a preferred module: cutters, stock gears and gauges exist for them. To convert, m = 25.4 ÷ DP; a gear of 10 DP has a module of 2.54 mm, which is not a standard module, and the two series will not mesh correctly.
Backlash and center distance
Backlash is the clearance between the non-driving faces of meshing teeth, needed for lubrication, thermal expansion and manufacturing errors. For general spur gears a backlash of about 0.04 to 0.08 times the module is common; precision and reversing drives (such as a servo axis, see Mechanical Transmission) need less, achieved with anti-backlash split gears, preloading or low-backlash gearboxes. Changing the center distance by Δa changes the backlash by about 2 Δa tan φ: opening the center distance by 0.1 mm adds roughly 0.073 mm of backlash. Mount the shafts accurately and give the housing the right tolerance (see Fits and Tolerances).
A checklist for choosing and checking a gear pair
- Ratio and speed: pick tooth counts so that i is as needed; avoid common factors between the two counts if the wear should spread evenly (a “hunting tooth” design).
- Module, face width and material: from the strength calculation in Gear Strength and Design Calculations.
- Teeth numbers: at least 17 for the pinion, or use a profile shift; check the contact ratio is above 1.4.
- Center distance: a = m (z1 + z2) ÷ 2, matched to the housing, with the shaft tolerances.
- Same module and pressure angle on both gears; right handedness for helical pairs (opposite hands for parallel shafts).
- Bearings, shafts and lubrication for the forces (see Shafts, Keys and Couplings, Bearings and Bearing Selection and Lubrication, Wear and Gear Failures).
Related
- Gears and Power Transmission: the overview of this guide.
- Gear Types and Selection, Gear Strength and Design Calculations and Gear Trains, Ratios and Efficiency: the next steps.
- Gear Ratio, Torque and Speed Generator.
- Fits and Tolerances for shaft and bore fits.
- Mechanical Transmission: gearboxes, backlash and inertia matching on servo axes.
FAQ
What is the difference between module and diametral pitch?
Module is the pitch diameter in millimeters divided by the number of teeth (m = d ÷ z): a larger module means a bigger tooth. Diametral pitch is the number of teeth per inch of pitch diameter (P = z ÷ d): a larger number means a smaller tooth. Convert with m = 25.4 ÷ P.
Can I mesh a gear of module 2 with one of module 2.5?
No. Gears must have the same module and pressure angle to mesh: with different modules the teeth have different pitches and cannot run together.
Why 20° pressure angle?
It gives a stronger tooth than 14.5° and a smaller minimum number of teeth (17 instead of 32). Most gears made today are 20°; 25° is used in some heavy-duty gears.
How many teeth should the pinion have?
At least 17 for a standard 20° tooth, 20 or more for a quiet and strong gear. Fewer teeth are possible with profile shift or for light loads, but strength and noise suffer.
References
- Shigley, J. E. and Mischke, C. R., Mechanical Engineering Design, McGraw-Hill: spur gears (chapter on gear geometry).
- ISO 53 (basic rack), ISO 54 (modules), ISO 21771 and DIN 3960 (gear terminology and geometry); AGMA 1012-G05 (gear nomenclature).
- KHK, Gear Technical Reference (free online): geometry, interference, backlash.
Last reviewed: October 2026. Values are for standard full-depth 20° involute teeth; use the gear standard for special tooth forms.