Every spur gear design starts with the same handful of numbers: pitch diameter, tooth depth, tip and root diameters, base circle, pitch, ratio, centre distance and contact ratio. Get these right and everything downstream, from strength checks to manufacturing drawings, has a solid foundation. This post walks through all eight calculations on one real gear pair, with every formula and every number shown, so you can follow along with a calculator.
Prefer to watch? The full worked example is in the video below (4½ minutes). The written version underneath adds a little more context on why each formula looks the way it does.
The gear pair we’ll calculate
We’ll use a common metric example: a 20-tooth pinion driving a 40-tooth gear.
| Input | Symbol | Value |
|---|---|---|
| Module | m | 4 mm |
| Pressure angle | α | 20° |
| Pinion teeth | z₁ | 20 |
| Gear teeth | z₂ | 40 |
| Pinion speed | n₁ | 1440 rpm |
Assumptions: standard full-depth involute teeth (addendum = 1.0 m, dedendum = 1.25 m), no profile shift, metric units. With 20 teeth the pinion is comfortably above the 17-tooth minimum that avoids undercut at a 20° pressure angle, so no correction is needed.
The four circles that define a spur gear
- Tip (addendum) circle: the outside of the teeth, and the size of the gear blank.
- Pitch circle: the imaginary circle on which the two gears roll together without slipping. Most gear dimensions are referenced to it.
- Base circle: the circle the involute tooth profile is generated from.
- Root (dedendum) circle: the bottom of the tooth spaces.
The addendum is the tooth height above the pitch circle; the dedendum is the depth below it.
Step 1: Pitch circle diameter
d = m × z
Pinion: d₁ = 4 × 20 = 80 mm
Gear: d₂ = 4 × 40 = 160 mmThe module is simply pitch diameter per tooth, which is why it is the one number every other dimension scales from. Two gears can only mesh if they share the same module and pressure angle.
Step 2: Addendum, dedendum and whole depth
ha = 1.0 × m = 4 mm
hf = 1.25 × m = 5 mm
h = ha + hf = 9 mmThe dedendum is 0.25 m deeper than the addendum. That extra quarter-module is the bottom clearance: it stops the tip of the mating tooth from bottoming out in the root and leaves room for the root fillet and lubricant.
Step 3: Tip and root diameters
da = d + 2m df = d − 2.5m
Pinion: da₁ = 80 + 8 = 88 mm df₁ = 80 − 10 = 70 mm
Gear: da₂ = 160 + 8 = 168 mm df₂ = 160 − 10 = 150 mmThe tip diameter is the outside diameter you turn the blank to before cutting teeth. The root diameter is useful for checking rim thickness and for keyway or bore clearance on small pinions.
Step 4: Base circle diameter
db = d × cos α (cos 20° = 0.9397)
Pinion: db₁ = 80 × 0.9397 = 75.18 mm
Gear: db₂ = 160 × 0.9397 = 150.35 mmThe involute flank of every tooth is generated from the base circle. Below it there is no involute, so the profile blends into the root fillet. On both of our gears the base circle sits just above the root circle (75.18 vs 70 mm on the pinion, 150.35 vs 150 mm on the gear). For standard 20° full-depth teeth that holds up to 41 teeth; from 42 teeth upward the base circle drops below the root circle and the entire working flank is involute.
Step 5: Circular pitch and tooth thickness
p = π × m = 3.1416 × 4 = 12.566 mm
s = p ÷ 2 = 6.283 mmCircular pitch is the distance from one tooth to the next, measured along the pitch circle. Both gears share it, which is exactly what lets them mesh. The theoretical tooth thickness is half the pitch; in practice a small amount is removed from each tooth to give backlash.
Step 6: Gear ratio and output speed
i = z₂ ÷ z₁ = 40 ÷ 20 = 2
n₂ = n₁ ÷ i = 1440 ÷ 2 = 720 rpmA ratio of 2 halves the speed and, ignoring losses, doubles the torque. Spur gear meshes are efficient (typically around 98–99% per stage), so the losses are small but not zero when you size a motor.
Step 7: Centre distance
a = (d₁ + d₂) ÷ 2 = (80 + 160) ÷ 2 = 120 mm
Check: a = m × (z₁ + z₂) ÷ 2 = 4 × 60 ÷ 2 = 120 mmThis is the theoretical shaft centre distance for standard gears. In the real housing, the centre distance tolerance together with the tooth thickness tolerance sets your backlash, so specify both together.
Step 8: Contact ratio check
Contact ratio tells you how many tooth pairs are in contact on average. It is the length of the path of contact divided by the base pitch:
Path of contact:
g = √(ra₁² − rb₁²) + √(ra₂² − rb₂²) − a × sin α
= √(44² − 37.59²) + √(84² − 75.18²) − 120 × sin 20°
= 22.873 + 37.479 − 41.042
= 19.31 mm
Base pitch:
pb = π × m × cos α = π × 4 × 0.9397 = 11.81 mm
Contact ratio:
ε = g ÷ pb = 19.31 ÷ 11.81 = 1.635Any value above 1 means a new pair of teeth engages before the previous pair lets go. A contact ratio of 1.2 or higher is a common design guideline; at 1.635 this pair runs smoothly, with two pairs of teeth sharing the load for about 64% of each mesh cycle.
Results at a glance
| Parameter | Pinion | Gear |
|---|---|---|
| Teeth, z | 20 | 40 |
| Pitch diameter, d | 80 mm | 160 mm |
| Tip diameter, da | 88 mm | 168 mm |
| Root diameter, df | 70 mm | 150 mm |
| Base diameter, db | 75.18 mm | 150.35 mm |
| Speed, n | 1440 rpm | 720 rpm |
Common to both gears: module 4 mm, pressure angle 20°, circular pitch 12.566 mm, tooth thickness 6.283 mm, whole depth 9 mm, centre distance 120 mm, gear ratio 2, contact ratio 1.635.
What this doesn’t cover
These eight steps fix the geometry. Before a gear pair is released for manufacture you still need to check:
- Tooth bending strength (Lewis equation for a first pass, then AGMA 2001 or ISO 6336)
- Surface contact (pitting) stress, which often governs hardened steel gears
- Material, heat treatment and gear quality grade
- Backlash, tolerances and lubrication, plus profile shift if you need a non-standard centre distance or fewer teeth
These calculations are for education and preliminary design. Need gears or a drivetrain checked properly, including FEA of the teeth, shafts and housing? Talk to our engineers, or browse our free engineering calculators.