Spur Gear Calculator: Dimensions and Formulas
Enter a module (or diametral pitch), the number of teeth and the pressure angle. The calculator returns every dimension you need to draw, machine, inspect and order a spur gear, including the involute cutter number that matches your tooth count.
Spur gear formulas at a glance
Standard full-depth involute gear, metric module system (m = module, z = teeth, α = pressure angle)
- Pitch (reference) diameter
- d = m × z
- Outside (tip) diameter
- da = m × (z + 2)
- Root diameter
- df = m × (z - 2.5)
- Base circle diameter
- db = m × z × cosα
- Addendum / dedendum
- ha = m ; hf = 1.25m
- Module from diametral pitch
- m = 25.4 / DP
Design details (mating gear, helix, profile shift, inspection)
Ordering details (for an accurate quote)
Cutting a range of tooth counts? Order the full set of 8 cutters to cover 12 teeth to a rack. See the involute gear cutters range.
Every spur gear is defined by three numbers: its tooth size, its tooth count and its pressure angle. Get those three right and every other dimension, the diameters, the tooth depth, the inspection figures and the cutter you order, follows from a handful of formulas. The calculator above runs all of them. The rest of this page explains where each number comes from, so you can check the output and use it with confidence in a drawing, a CNC setup or a purchase order.
The two sizing systems: module and diametral pitch
Tooth size is specified one of two ways, and the only real difference is the unit. The module (used across Europe, India and most of Asia) is the pitch diameter in millimetres divided by the number of teeth, so a larger module means a larger tooth. Diametral pitch or DP (used in North America and legacy British work) is the number of teeth per inch of pitch diameter, so a larger DP means a smaller tooth. They are inverse, linked through the 25.4 mm in an inch:
m = 25.4 / DP and DP = 25.4 / m
So a 2 mm module is about 12.7 DP, and a common 24 DP gear is about a 1.058 module. The number you must never mix up is the pressure angle, the angle of the tooth flank at the pitch line. The world standard is 20 degrees. You will still meet 14.5 degrees on older equipment and 25 degrees where extra tooth strength is needed. A cutter ground for one pressure angle will not produce a correct tooth at another, so this value carries through every calculation below.
Spur gear formula reference
These are the standard relationships for a full-depth involute tooth (the proportions defined by the ISO 53 and DIN 867 basic rack). Both unit systems are shown side by side.
| Parameter | Module (metric) | Diametral pitch (imperial) |
|---|---|---|
| Pitch (reference) diameter | d = m × z | D = N / P |
| Base circle diameter | db = m × z × cosα | Db = D × cosα |
| Addendum | ha = m | a = 1 / P |
| Dedendum | hf = 1.25 × m | b = 1.25 / P |
| Whole depth | h = 2.25 × m | h = 2.25 / P |
| Working depth | hk = 2 × m | hk = 2 / P |
| Clearance | c = 0.25 × m | c = 0.25 / P |
| Outside (tip) diameter | da = m × (z + 2) | Do = (N + 2) / P |
| Root diameter | df = m × (z - 2.5) | Dr = (N - 2.5) / P |
| Circular pitch | p = π × m | pc = π / P |
| Tooth thickness (at pitch line) | s = π × m / 2 | t = π / (2 × P) |
| Centre distance (pair) | a = m × (z1 + z2) / 2 | C = (N1 + N2) / (2 × P) |
Worked example: metric gear
Take a 2 module gear with 20 teeth and a 20 degree pressure angle.
- Pitch diameter. Multiply module by teeth.d = 2 × 20 = 40.000 mm
- Outside diameter. Add two modules.da = 2 × (20 + 2) = 44.000 mm
- Root diameter. Subtract 2.5 modules.df = 2 × (20 - 2.5) = 35.000 mm
- Base circle. Pitch diameter times cosine of the pressure angle.db = 40 × cos 20 = 37.588 mm
- Tooth depth. Addendum 2.000 mm, dedendum 2.500 mm, whole depth 4.500 mm.
That 20 tooth count places the gear in the range of a No. 6 involute cutter, which is the figure the calculator returns automatically.
Worked example: imperial gear
Take a 24 DP gear with 40 teeth at 20 degrees.
- Pitch diameter. Teeth divided by DP.D = 40 / 24 = 1.6667 in
- Outside diameter. Teeth plus two, divided by DP.Do = 42 / 24 = 1.7500 in
- Root diameter.Dr = 37.5 / 24 = 1.5625 in
- Module equivalent, if you order a metric cutter.m = 25.4 / 24 = 1.0583 mm
Tooth proportions by pressure angle
The standard proportions are constant in modules, but the practical minimum tooth count changes with pressure angle, because a flatter flank (lower angle) needs more teeth before the involute clears the base circle.
| Pressure angle | Addendum | Dedendum | Whole depth | Min. teeth (no undercut) |
|---|---|---|---|---|
| 14.5° | 1.000 m | 1.157 m | 2.157 m | about 32 |
| 20° (standard) | 1.000 m | 1.250 m | 2.250 m | about 17 to 18 |
| 25° | 1.000 m | 1.250 m | 2.250 m | about 12 |
Undercut and the minimum tooth count
When a gear has too few teeth, the generating cutter cuts into the flank below the base circle and removes part of the working profile. This is undercut, and it weakens the tooth root and shortens the contact. The theoretical limit is:
z(min) = 2 / sin²α
For a 20 degree pressure angle that is 17.1, so 17 teeth shows slight undercut and 18 is fully clear. Below those counts you have two options: switch to a higher pressure angle, or apply a positive profile shift (set the x value in the calculator above), which moves the cutter outward and rebuilds the root. The calculator flags any tooth count that falls under the limit for the angle you chose.
From gear dimensions to the right cutter
A single involute milling cutter does not cut every gear. Because the tooth shape changes with tooth count, a standard set splits the range into eight cutters, numbered 1 (for the largest gears, up to a rack) through 8 (for the smallest, 12 to 13 teeth). The calculator maps your tooth count to the correct number, but the full table is below for reference.
| Cutter No. | Tooth range it cuts |
|---|---|
| 1 | 135 teeth to a rack |
| 2 | 55 to 134 |
| 3 | 35 to 54 |
| 4 | 26 to 34 |
| 5 | 21 to 25 |
| 6 | 17 to 20 |
| 7 | 14 to 16 |
| 8 | 12 to 13 |
One caution worth knowing before you buy: many imported sets print the numbers in reverse order, so a cutter marked No. 1 is actually for the smallest gears. Always read the engraved tooth range, not just the number. For the full selection logic, including module versus DP marking and pressure angle, see our guide on how to choose an involute gear cutter, and the specifications on the involute gear cutters page.
Inspection: span measurement and measurement over pins
Calculated dimensions are only half the job. To verify a finished gear you need two figures the calculator also returns.
The base tangent length, or span W, is the distance a disc micrometer reads across several teeth. It is the cleanest check of tooth thickness because the reading is independent of where the anvils touch the flank. The standard relationship (DIN 3960, ISO 21771) is:
W = m × cosα × [ π × (k - 0.5) + z × invα ]
where invα = tanα - α (the involute function, angle in radians) and k is the number of teeth to span, taken as the nearest whole number to z × α / 180 + 0.5.
Measurement over pins (or balls) M places two gauge pins in opposite tooth spaces and measures across them. It needs the involute solved in reverse, which the calculator does numerically. Use the best pin size for a clean contact near the pitch line: about 1.728m for 20 degrees, 1.901m for 14.5 degrees and 1.616m for 25 degrees. Override the value in the calculator if your gauge set is fixed.
How to size a replacement gear or cutter from a sample
A common job is reverse engineering a worn gear with no drawing. The fast method:
- Count the teeth (z).
- Measure the outside diameter (da) with a caliper.
- Estimate the module: m ≈ da / (z + 2), then round to the nearest standard module (0.5, 0.75, 1, 1.25, 1.5, 2, 2.5, 3 and so on).Example: da = 44 mm, z = 20, so m = 44 / 22 = 2.0
- Confirm the pressure angle (almost always 20 degrees) and verify with a base tangent span reading.
Enter those values above and you have the full dimension set plus the cutter number to order. If the result lands between two standard modules, the original was probably a DP gear: try DP = (z + 2) / da(in) instead.
Gear blank sizing for the shop floor
Before any teeth are cut, the blank is turned to the outside diameter, da = m × (z + 2). Oversizing wastes material and machining time on hardened or alloy stock; undersizing leaves incomplete tooth tips, which is unrecoverable. Add only your normal finishing allowance to the calculated OD, keep the bore and faces square to the axis (runout transfers straight into tooth-to-tooth error), and you are ready to index and cut.
Frequently asked questions
How do I calculate the module of a spur gear?
The module equals the pitch diameter divided by the number of teeth: m = d / z. If you only have the outside diameter of an existing gear, estimate it with m is approximately equal to OD / (z + 2), then confirm by checking the result against a standard module.
What is the formula for the outside (tip) diameter of a spur gear?
For a standard full-depth gear the outside diameter is the module multiplied by the number of teeth plus two: da = m × (z + 2). In the imperial diametral pitch system it is (N + 2) / P.
How do I find the centre distance between two spur gears?
For standard gears with no profile shift, the centre distance is half the module times the sum of the tooth counts: a = m × (z1 + z2) / 2. In the imperial system it is (N1 + N2) / (2 × P).
How do I convert module to diametral pitch?
Module and diametral pitch are inversely related through 25.4: module = 25.4 / DP, and DP = 25.4 / module. A 2 mm module is roughly 12.7 DP.
What is the minimum number of teeth to avoid undercut?
The theoretical minimum is 2 divided by the square of the sine of the pressure angle. That gives about 17 to 18 teeth for a 20 degree pressure angle, about 32 teeth for 14.5 degrees, and about 12 teeth for 25 degrees. Positive profile shift lets you cut fewer teeth without undercut.
How do I measure the module of an unknown gear?
Count the teeth, measure the outside diameter with a caliper, then estimate the module as OD / (z + 2). Round to the nearest standard module and confirm with a base tangent span measurement. This is the usual way to size a replacement gear or cutter.
Need the cutter for your gear?
Maxwell Tools has manufactured HSS gear cutting tools since 1976, exporting to more than 30 countries. Send us your module or DP, tooth count and pressure angle, or a sample, and we will quote the exact involute cutter, full set or custom profile you need.