This pitch diameter calculator works out the pitch diameter of toothed power-transmission components — spur gears in either the module or the diametral pitch system, roller chain sprockets, and synchronous belt pulleys. That is one of two completely different things the phrase "pitch diameter" means in engineering, and it is worth being clear which one you are after before you read any further.
The other meaning belongs to screw threads, where the pitch diameter is the diameter of an imaginary cylinder cutting the thread where groove and ridge widths are equal, and it is the dimension a thread gauge actually measures. If that is what you came for, our thread pitch calculator derives it along with minor diameter and thread depth from a designation like M10×1.5, and the clearance hole calculator covers hole and tap drill sizing. Arb Digital keeps them as separate tools because they share a name and share nothing else.
What This Pitch Diameter Calculator Does
The pitch diameter of a toothed component is the diameter of the imaginary circle that rolls without slipping against its mate. Gears do not really touch on that circle at every instant, and a chain certainly does not roll on it, but it is the reference every other dimension is measured from, and it is what sets ratio and centre distance. Nothing about a gear train makes sense until you have it.
Give the calculator a tooth count and a pitch and it returns the pitch diameter, the tip or outside diameter, the circular pitch measured along the pitch circle, the base or bottom diameter, and the theoretical centre distance to a mating component of the tooth count you specify. In gear modes it also converts between module and diametral pitch, so you can check whether a metric gear and an inch gear will run together — usually the answer is that they will not.
This is a geometry tool. It does not rate a gear, a chain or a belt for torque, power or life. Those are capacity questions answered by the manufacturer's rating tables with service factors applied, and a mesh that is geometrically correct can still be badly undersized. If you need ratio, output speed and torque across a train, the gear ratio calculator does that, and the pulley calculator handles the speed and diameter relationship for plain and V-belt drives.
How to Use It
- Choose the component type. Gears split into the metric module system and the inch diametral pitch system; the two are not interchangeable.
- Enter the number of teeth on the component you are working out, and the teeth on its mate if you want a centre distance.
- Enter the pitch — module in millimetres, diametral pitch in teeth per inch, or the chain or belt tooth pitch.
- Set the pressure angle for gears. Twenty degrees is by far the most common modern value, but 14.5 and 25 degree systems exist and a gear will not mesh correctly with a different pressure angle.
- For sprockets and timing pulleys, enter the roller diameter or pitch line differential from the chain or belt standard to get a meaningful bottom or outside diameter.
The Formula and How It Is Calculated
Metric spur gear. The pitch diameter is simply d = m × z. A module 2 gear with 40 teeth has a pitch diameter of 80 mm. The tip diameter adds one module of addendum at each end, da = m(z + 2) = 84 mm. The root diameter subtracts a dedendum of 1.25 m at each end, df = m(z − 2.5) = 75 mm. Circular pitch is p = πm = 6.283 mm, and base diameter is db = d·cos α = 80 × cos 20° = 75.18 mm.
Inch spur gear. The relations mirror the metric ones with the pitch inverted: d = z ÷ P, da = (z + 2) ÷ P, circular pitch = π ÷ P. Module and diametral pitch convert as m = 25.4 ÷ P, so a 12 diametral pitch gear corresponds to a module of about 2.117 mm — close to module 2 but not equal to it, which is exactly why the two systems do not mesh.
Roller chain sprocket. The teeth sit on a polygon rather than a circle, so the pitch diameter comes from the chord relation PD = p ÷ sin(180° ÷ z). An ANSI 40 chain of half-inch pitch on a 17-tooth sprocket gives 0.5 ÷ sin 10.588° = 2.721 inches. The bottom diameter, which is what a caliper reads across the seated rollers, is the pitch diameter minus twice the roller diameter.
Synchronous belt pulley. A toothed belt does wrap a true circle, so PD = (z × p) ÷ π. Thirty teeth at 5 mm pitch gives 47.75 mm. The outside diameter is smaller than the pitch diameter, not larger, because the belt's pitch line sits above the tooth face: OD = PD − 2 × PLD, using the pitch line differential the belt manufacturer publishes for that tooth profile.
The Polygon Effect, and Why Sprockets Are Not Gears
The sine in the sprocket formula is not a decoration. A chain is a series of rigid links, so it wraps a sprocket as a polygon, and the effective radius the chain runs at changes as each link engages. On a sprocket with few teeth this produces a real cyclic variation in chain speed and tension called chordal action, and it gets worse fast as the tooth count falls — it is the main reason drive designers avoid very small sprockets even when the ratio would allow one.
The same effect explains why the pitch diameter of a sprocket is not a diameter you can measure. There is no material at the pitch circle; it passes through the centres of the chain pins. That is why the bottom diameter is given as well, because it is the dimension you can actually get a caliper across on a real sprocket.
Centre Distance Is a Starting Point, Not an Answer
For a pair of external gears the theoretical centre distance is half the sum of the two pitch diameters, and the calculator reports it that way. Real installations depart from it deliberately and often. Gears are frequently cut with profile shift, which moves the operating pitch diameter away from the nominal one so that a pair can run at a non-standard centre distance or so that a small pinion avoids undercutting. Backlash has to be allowed for, and it is normally created by cutting the teeth slightly thin rather than by opening the centres.
For chain drives the centre distance is not fixed by the sprockets at all — it is set by the chain length in whole pitches, and it usually must be adjustable because chain elongates in service. For synchronous belts the centre distance is fixed by the belt's tooth count, and the belt manufacturer publishes the exact relation along with the tension the drive requires. Treat the figure this page gives as the first line of a design, not the last.
Module, Diametral Pitch and the Mistake That Ruins a Pair
Module and diametral pitch describe the same idea from opposite ends. Module is the millimetres of pitch diameter contributed by each tooth, so a bigger module means bigger teeth. Diametral pitch is the number of teeth per inch of pitch diameter, so a bigger number means smaller teeth. People coming from one system regularly read the other backwards.
They are also almost never interchangeable in practice. Module 2 and 12 diametral pitch are within a few per cent of each other, which is close enough to look right on a drawing and far enough apart to mesh badly, wear quickly and run noisily. Two gears mesh correctly only if they share the same module or diametral pitch and the same pressure angle. The calculator prints the equivalent in the other system precisely so you can see how close a near-miss is.
The standards behind these definitions are published rather than invented here. Gear terminology and tooth proportions are covered by the ANSI/AGMA standards developed through ANSI-accredited committees, and internationally by ISO 53, Cylindrical gears for general and heavy engineering — Standard basic rack tooth profile. Roller chain and sprocket dimensions are defined in ASME B29.1 for ANSI chain and ISO 606 for metric chain; the ASME codes and standards catalogue is the index for the former. Look up the standard for the specific series you are using rather than relying on any single-page summary.
Arb Digital builds websites and technical search visibility for manufacturers and component distributors. Specifiers search by part and by capability — your catalogue should be what they land on.
SEO Services Web Design ServicesCommon Mistakes to Avoid
- Confusing gear pitch diameter with thread pitch diameter. They share a name and nothing else; a screw thread's pitch diameter is a gauging dimension on a helix, not a rolling circle.
- Measuring a gear's outside diameter and calling it the pitch diameter. The tip diameter is two addenda larger, and on a low tooth count that is a large proportional difference.
- Assuming module 2 and 12 diametral pitch will mesh. They are close enough to fit and far enough apart to fail; matching pitch and pressure angle are both required.
- Trying to caliper a sprocket's pitch diameter. There is no material there — it passes through the chain pin centres. Measure across the seated rollers and work back.
- Expecting a timing pulley's outside diameter to be larger than its pitch diameter. It is smaller, by twice the pitch line differential, because the belt's pitch line sits above the pulley teeth.
Related Free Tools From Arb Digital
Work ratio, output speed and torque with the gear ratio calculator, handle plain and V-belt drives with the pulley calculator, derive screw thread geometry with the thread pitch calculator, size fastener holes with the clearance hole calculator, and set out hole patterns with the bolt circle calculator. Browse the full free online tools hub, or contact us if a calculator you need is missing.
Frequently Asked Questions
For a toothed component it is the diameter of the imaginary circle that rolls without slipping against its mate. It is the reference every other tooth dimension is measured from, and it is what determines ratio and centre distance.
No. On a screw thread, pitch diameter is the diameter of an imaginary cylinder that cuts the thread where the groove and ridge widths are equal, and it is a gauging dimension. The two share a name and nothing else. Use the thread pitch calculator for threads.
In the metric system multiply the module by the number of teeth. In the inch system divide the number of teeth by the diametral pitch. A module 2 gear with 40 teeth has an 80 mm pitch diameter.
Because chain links are rigid, so the chain wraps the sprocket as a polygon rather than a circle. The pitch diameter is the chord relation, pitch divided by the sine of 180 degrees over the tooth count, and the difference matters most on low tooth counts.
Generally no. Module 2 and 12 diametral pitch are within a few per cent, which is close enough to assemble and far enough apart to mesh badly and wear quickly. Correct meshing needs the same pitch and the same pressure angle.
Because the belt's pitch line runs through the tensile cords above the tooth face, so the pitch circle sits outside the pulley material. The outside diameter is the pitch diameter minus twice the pitch line differential published for that belt profile.
No. It is geometry only. Torque, power and life ratings come from the gear, chain or belt manufacturer's tables with the appropriate service factor applied, and a geometrically correct mesh can still be badly undersized.
This page computes published geometric relations from the values you enter, for design study and education only. It publishes no rating data and it certifies nothing. Component selection, load rating, service factors, lubrication and guarding must follow the manufacturer's data and the applicable standards, and where a drive matters the design should be checked by a qualified engineer.
