Pitch Diameter Calculator
Work out the pitch diameter and geometry of a spur gear. Enter the number of teeth, the pressure angle and the pitch (diametral pitch, module or circular pitch), and get the pitch diameter, outside and root diameters, base circle, circular pitch and tooth thickness — plus the gear ratio and center distance for a mating gear. Includes by-teeth and by-pitch tables and a downloadable PDF.
| Teeth | Pitch diameter | Outside dia. | Base circle |
|---|---|---|---|
| Enter a gear to compare | |||
| Pitch | Pitch diameter | Outside dia. | Circular pitch |
|---|---|---|---|
| Enter a gear to compare | |||
How to use the pitch diameter calculator
Four quick steps turn a gear’s teeth and pitch into its full geometry — pitch, outside and root diameters, base circle, circular pitch and tooth thickness — plus the ratio and center distance for a mating gear, with by-teeth and by-pitch tables and a downloadable report.
Enter the gear
The number of teeth and the pressure angle (14.5°, 20° or 25°).
Pick the pitch system
Diametral pitch, module or circular pitch, then enter the pitch value.
Add a mating gear
Optionally enter the mate’s teeth for the ratio and center distance.
Read results & save a PDF
Get the pitch diameter and full tooth geometry in inches and mm — then download a report.
The formulas behind the geometry
Pitch diameter follows from the teeth and the pitch; from it the whole standard full-depth involute tooth geometry falls out, along with the mating-gear ratio and center distance.
▶ Worked example
A 24-tooth spur gear at diametral pitch 10, 20° pressure angle, meshing with a 48-tooth gear.
Pitch diameter is 24 ÷ 10 = 2.4 in (60.96 mm); addendum 0.1, so OD is 2.4 + 0.2 = 2.6 in and root 2.15 in.
Base circle = 2.4 × cos 20° ≈ 2.255 in · circular pitch 0.3142 in
The 48-tooth mate has a 4.8 in pitch diameter, so the center distance is (2.4 + 4.8) ÷ 2 = 3.6 in and the ratio is 2:1.
Gear geometry quick reference
The pitch diameter formula for each pitch system, the full standard tooth geometry, and the pitch diameter of common tooth counts at DP 10.
| System | Pitch diameter | Units | Addendum |
|---|---|---|---|
| Diametral pitch (DP) | teeth ÷ DP | inches | 1 ÷ DP |
| Module (m) | teeth × module | mm | module |
| Circular pitch (CP) | teeth × CP ÷ π | inches | CP ÷ π |
Both gears in a mesh must share the same pitch system and value.
| Quantity | Formula |
|---|---|
| Pitch diameter (PD) | by system (above) |
| Addendum (a) | 1 ÷ DP, module, or CP ÷ π |
| Dedendum (d) | 1.25 × a |
| Outside diameter | PD + 2a |
| Root diameter | PD − 2d |
| Whole depth | a + d |
| Base circle | PD × cos(pressure angle) |
| Circular pitch | π × PD ÷ teeth |
| Tooth thickness | circular pitch ÷ 2 |
| Teeth | Pitch diameter | Outside diameter | Base circle |
|---|---|---|---|
| 12 | 1.200 in | 1.400 in | 1.128 in |
| 18 | 1.800 in | 2.000 in | 1.691 in |
| 24 | 2.400 in | 2.600 in | 2.255 in |
| 36 | 3.600 in | 3.800 in | 3.383 in |
| 48 | 4.800 in | 5.000 in | 4.510 in |
| 72 | 7.200 in | 7.400 in | 6.766 in |
At a different diametral pitch, scale the pitch diameter as teeth ÷ DP. Enter your own figures above for any system.
Everything the calculator gives you
One gear produces a full geometry profile, not just a single number.
Gear geometry reference numbers
Built for anyone working with gears
Machinists, designers, and hobbyists all need a gear’s pitch diameter and tooth geometry.
Cutting a spur gear and needing the blank diameter and tooth depth. The outside diameter sets the blank; the whole depth sets how deep to cut.
- Get the outside diameter to turn the blank to
- Read the whole depth for the cutter setting
- Confirm the pitch and pressure angle of the cutter
Laying out a gear train — the ratio, the center distance between shafts, and the pitch diameters that have to fit the housing.
- Work out the ratio from the tooth counts
- Get the center distance for the shaft layout
- Compare pitch diameters in inch and metric
Identifying or replacing a gear in a machine, clock or model, and reverse-engineering its geometry from the teeth and pitch.
- Check a replacement matches the original pitch
- Work out the geometry from a measured gear
- Convert between diametral pitch and module
Get the gear geometry right
Small habits make a gear layout far more reliable.
Match pitch and pressure angle
Two gears mesh only if they share both. Different pitches will not interlock, whatever the tooth count.
Use 20° by default
20° is the modern standard. Only use 14.5° to match older gears, or 25° for stronger teeth.
Turn the blank to OD
The outside diameter, not the pitch diameter, is what you machine the gear blank to.
Mind the system
Diametral pitch and circular pitch are inch; module is metric. Don’t mix them in one mesh.
Center distance from PDs
At the same pitch, the shaft spacing is half the sum of the two pitch diameters.
Standard full-depth only
These relations are for standard spur gears. Stub, helical, profile-shifted and bevel gears differ.
Confirm before cutting
Check the figures against the gear standard and manufacturer data before committing to a blank.
Frequently asked questions
Common questions about pitch diameter, pitch systems, and gear geometry.
The pitch diameter is the diameter of a gear’s pitch circle — the imaginary circle where two meshing gears effectively roll against each other without slipping. It is the single most important dimension for laying out a gear and a gear train.
It is not a physical edge you can measure directly; it sits partway up the teeth. The teeth extend outward to the outside diameter and inward to the root, with the pitch circle between them.
Enter the number of teeth, the pressure angle and the pitch, and this calculator returns the pitch diameter in inches and millimeters, along with the full tooth geometry.
Pitch diameter comes from the number of teeth and the pitch, and the formula depends on the pitch system. In the inch (diametral pitch) system it is the number of teeth divided by the diametral pitch; in metric it is the number of teeth times the module; with circular pitch it is the teeth times the circular pitch divided by pi.
For example, a 24-tooth gear at a diametral pitch of 10 has a pitch diameter of 24 divided by 10, which is 2.4 inches, or about 61 millimeters.
So the more teeth at a given pitch, the larger the gear. This calculator does the math for whichever pitch system you choose and shows the result in both inches and millimeters.
These are three ways to express how fine or coarse a gear’s teeth are, and every gear in a mesh must share the same one. Diametral pitch (DP) is the number of teeth per inch of pitch diameter, used in the inch system — a higher DP means finer teeth.
Module is the metric measure: the millimeters of pitch diameter per tooth, so a larger module means bigger teeth. Circular pitch is the distance from one tooth to the next measured along the pitch circle.
They are related: module equals 25.4 divided by the diametral pitch, and circular pitch equals pi divided by the diametral pitch. This calculator accepts any of the three and converts the geometry accordingly.
The pitch diameter is where the gears mesh; the outside diameter is the overall diameter across the tips of the teeth, which is what you actually turn a gear blank to. The outside diameter is larger by twice the addendum — the height of a tooth above the pitch circle.
For a standard full-depth gear the addendum is one divided by the diametral pitch (or one module in metric), so the outside diameter is the pitch diameter plus two addendums.
For example, a 24-tooth gear at diametral pitch 10 has a pitch diameter of 2.4 inches and an outside diameter of 2.6 inches. This calculator reports both, plus the root diameter at the bottom of the teeth.
The center distance is how far apart the two shaft centers sit when the gears mesh. For standard gears it is simply the sum of the two pitch diameters divided by two, which is also the average of the two pitch radii.
Both gears must be at the same pitch for this to hold. A larger gear and a smaller gear at the same pitch will mesh at a center distance set by their combined size.
For example, a 24-tooth and a 48-tooth gear, both at diametral pitch 10, have pitch diameters of 2.4 and 4.8 inches, so the center distance is 3.6 inches. This calculator works out the center distance and the gear ratio when you enter a mating gear.
The pressure angle is the angle of the tooth flank where the teeth make contact, measured from a line tangent to the pitch circles. The common standard is 20 degrees; 14.5 degrees is an older standard and 25 degrees gives stronger teeth with more load capacity.
It affects the shape of the tooth and the forces in the mesh, and it sets the base circle, from which the involute tooth profile is generated. A higher pressure angle makes a stubbier, stronger tooth but increases the separating force between the gears.
Both meshing gears must share the same pressure angle. This calculator offers 14.5, 20 and 25 degrees and uses the value to work out the base circle.
The base circle is the circle from which the involute curve of the tooth is generated — the tooth profile is the path traced by unwinding a taut string from this circle. It lies inside the pitch circle.
Its diameter is the pitch diameter times the cosine of the pressure angle. A larger pressure angle gives a smaller base circle relative to the pitch circle, which changes the tooth shape.
For example, a 2.4-inch pitch diameter at a 20-degree pressure angle has a base circle of 2.4 times the cosine of 20 degrees, about 2.255 inches. This calculator reports the base circle for your gear.
No — two gears can only mesh properly if they share the same pitch and the same pressure angle. The pitch sets the tooth size and spacing, so gears of different pitch have teeth that simply will not interlock correctly.
You can combine any numbers of teeth to get the ratio you want, as long as both gears are cut to the same pitch and pressure angle. The ratio is just the driven gear’s teeth divided by the driver’s teeth.
If you are matching a replacement to an existing gear, measure or identify its pitch and pressure angle first. This calculator assumes both gears share the pitch you enter and reports their ratio and center distance.
Disclaimer: Inch and metric. Pitch diameter = teeth ÷ diametral pitch (inch) = teeth × module (mm) = teeth × circular pitch ÷ π (inch). Values are for a standard full-depth involute spur gear (addendum = 1/DP, dedendum = 1.25/DP); base circle = pitch diameter × cos(pressure angle). Profile-shifted, stub, helical, bevel and worm gears — and screw-thread or pulley pitch diameters — follow different relations. Meshing gears must share the same pitch and pressure angle; confirm against the gear standard and manufacturer data before cutting.