Pitch Diameter Calculator uses PD = N / DP for spur gears, PD = N × m for metric gears, d2 = d − 0.6495P for threads, P / sin(π/N) for sprockets, and NP / π for timing pulley pitch diameter results.
This Pitch Diameter Calculator computes the theoretical pitch diameter for spur gears, threads and screws, roller chain sprockets, and timing pulleys in both US Customary and metric units. Select your component type, enter the two defining parameters, and the calculator returns the theoretical pitch diameter along with supporting geometry cards.
What this pitch diameter calculator does
The calculator takes a component type, a measurement system, and two input parameters, then returns the theoretical pitch diameter as the primary result. Supporting result cards show related geometry derived directly from that pitch diameter value.
The scope is exact. This is a theoretical geometry calculator. It does not calculate manufacturing tolerances, thread fit class limits, full sprocket tooth profile geometry, or timing pulley outside diameter. Every output is a calculated theoretical value based on the formulas described below.
The result changes by component type because each type uses a different pitch diameter formula. Switching the measurement system relabels inputs and converts outputs — the underlying geometry does not change.
Pitch diameter formulas used by the calculator
Spur Gear
US Customary — divide tooth count by diametral pitch:
Metric — multiply tooth count by module:
Diametral pitch DP is teeth per inch of pitch diameter. Module m is pitch diameter divided by tooth count in mm. They are reciprocals after unit conversion: m = 25.4 / DP.
Thread / Screw
For a US thread, convert TPI to pitch first:
Then apply the basic pitch diameter formula:
For metric threads, P is entered directly in mm. This formula gives the theoretical pitch cylinder where thread ridge width equals groove width.
Roller Chain Sprocket
Pitch diameter runs through the chain roller centers:
Approximate outside diameter:
This OD approximation ensures chain plates clear the tooth tips. It is not a full tooth profile calculation.
Timing Pulley
With tooth count N and belt pitch P:
For timing pulleys, the pitch line is embedded inside the belt. The physical outside diameter is always smaller than PD by an amount that depends on belt pitch-line distance and tooth profile. This calculator gives PD only — not pulley OD.
How to use the pitch diameter calculator
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Select Component Type Choose Spur Gear, Thread / Screw, Roller Chain Sprocket, or Timing Pulley. This determines which pitch diameter formula runs and which input labels appear.
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Select Measurement System US Customary uses inches, diametral pitch, and TPI. Metric uses millimetres, module, and thread pitch in mm. The unit system affects labels, defaults, and the Alternate Unit Eq. card output direction.
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Enter the two parameters For a spur gear: Number of Teeth and Diametral Pitch (or Module). For a thread: Major Diameter and TPI (or Thread Pitch). For a sprocket or pulley: Number of Teeth and Pitch Distance.
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Read the Theoretical Pitch Diameter first This is the primary output in the hero card — the working pitch circle or pitch cylinder for the selected component.
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Use the supporting result cards for related geometry Each card expands on the pitch diameter with additional calculated dimensions such as outside diameter, addendum, circular pitch, and unit conversions.
| Input | Applies To | What it means |
|---|---|---|
| Component Type | All | Selects whether the pitch diameter formula applies to a spur gear, thread, sprocket, or timing pulley. Each type uses a different formula and different supporting calculations. |
| Measurement System | All | Switches between US Customary (inches, DP, TPI) and Metric (mm, module, pitch in mm). Also controls the Alternate Unit Eq. card output direction. |
| N | Gear, Sprocket, Pulley | Number of teeth on the gear, sprocket, or pulley. Used directly in the numerator or multiplier of the pitch diameter formula. |
| DP | Gear — US | Diametral pitch: teeth per inch of pitch diameter. Higher DP means finer, smaller teeth. Defined as DP = N / PD. |
| m | Gear — Metric | Module: pitch diameter divided by tooth count, in mm. Larger module means larger, coarser teeth. Related to DP by m = 25.4 / DP. |
| Major Diameter | Thread | The outer diameter of the thread before subtracting thread depth geometry. For a ¼-20 thread, the major diameter is 0.250 in. |
| TPI | Thread — US | Threads per inch along the fastener axis. The calculator converts this to pitch P = 1/TPI before applying the pitch diameter formula. |
| Thread Pitch (P) | Thread — Metric | Axial distance between adjacent thread crests in mm. Entered directly for metric threads. |
| Pitch Distance | Sprocket, Pulley | Center-to-center distance between repeating features: chain roller centers for sprockets, belt tooth centers for pulleys. Standard ANSI chain pitches include 0.375 in (#35), 0.500 in (#40/#41), and 0.625 in (#50). |
Spur gear pitch diameter example
Inputs: 24 teeth, 8 DP, US Customary
With N = 24 teeth and DP = 8 (1/in), the pitch diameter formula gives:
The calculator then derives all supporting card values from this 3.000 in pitch diameter:
- Outside Diameter (OD)3.250 in
- Addendum (a = 1/DP)0.125 in
- Dedendum (1.157/DP)0.145 in
- Circular Pitch (CP)0.393 in
- Tooth Thickness0.196 in
- Working Depth0.250 in
- Travel per Revolution9.425 in
- Pitch Radius (r)1.500 in
- Base Circle (20° PA)2.819 in
- Metric Equiv. PD76.200 mm
- Equiv. OD82.550 mm
- Equiv. Module (m)3.175 mm
OD = (N+2)/DP. Addendum a = 1/DP = 0.125 in. Dedendum uses the standard full-depth constant: 1.157/DP = 0.145 in. Circular pitch CP = π/DP = 0.393 in. Tooth thickness t = CP/2. Working depth WD = 2a. Travel per revolution C = π × PD. Base circle BC = PD × cos(20°) = 2.819 in.
What the result cards mean
Theoretical Pitch Diameter
The main result shown in the hero card. This is the diameter of the working pitch circle for gears, sprockets, and pulleys, or the pitch cylinder for threads — the theoretical surface where motion or load transfer occurs. All other result cards derive from this value.
Outer Geometry
Shown for spur gears. Primary value is outside diameter OD = (N+2)/DP — the tip-to-tip measurement across the gear teeth. Sub-rows show addendum a = 1/DP (tooth height above the pitch circle) and dedendum b = 1.157/DP (tooth depth below). These are theoretical full-depth values, not inspection tolerances.
Circular Pitch
Shown for spur gears. Circular pitch CP = π/DP is the arc length from one tooth center to the next along the pitch circle. Sub-rows give tooth thickness t = CP/2 and working depth WD = 2a — the total engagement depth of mating teeth at full depth.
Travel per Revolution
Shown for spur gears. Primary value is pitch circumference C = π × PD — the linear distance the pitch circle covers in one full rotation. Sub-rows show pitch radius r = PD/2 and base circle diameter BC = PD × cos(20°) at the standard 20° pressure angle used in involute tooth geometry.
Alternate Unit Eq.
Shown for spur gears and threads. Converts the theoretical pitch diameter and key related dimensions into the opposite unit system. For a US Customary gear: metric equivalent PD in mm, equivalent OD in mm, and equivalent module. For a metric gear: inch equivalents and equivalent DP. The conversion is a direct unit change — no geometry is altered.
Thread / Screw Results
Hero card gives basic pitch diameter d2 = d − 0.6495P. Card 1 shows thread pitch and lead. Card 2 shows external basic minor diameter d3 = d − 1.22687P and external thread depth h3 = 0.61343P. Card 3 shows internal basic minor diameter D1 = d − 1.08253P and internal thread depth H1 = 0.54127P. Card 4 shows the alternate unit conversion.
Sprocket Results
Hero card gives pitch diameter through the chain roller centers: PD = P / sin(π/N). Card 1 shows approximate outside diameter and pitch radius r = PD/2. The OD approximation ensures chain plates clear tooth tips — it is not a full tooth profile. Card 2 gives chain travel per revolution = N × P. Card 3 converts to the opposite unit system.
Timing Pulley Results
Hero card gives pitch diameter PD = (N × P)/π. Card 1 shows pitch radius r = PD/2. Card 2 gives belt travel per revolution = N × P. Card 3 converts to the opposite unit system. This calculator does not output pulley OD — the physical outside diameter is always smaller than PD by a belt-dependent offset requiring belt pitch-line distance and tooth profile data.
When pitch diameter is useful
- Gear sizing and mesh checks Pitch diameter determines the center distance between two meshing gears: C = (PD1 + PD2) / 2. Two gears with the same DP (or module) and the correct center distance will mesh correctly when their pitch circles are tangent.
- Thread geometry estimates The basic pitch diameter d2 is the starting point for checking thread fit and engagement length. It gives the theoretical mid-flank diameter before applying tolerance class limits from ANSI, ISO, or ASME thread standards.
- Sprocket chain path estimates Sprocket pitch diameter — through the chain roller centers — lets you estimate chain wrap arc length, center distance between two sprockets, and chain length in links for a given drive configuration.
- Timing pulley belt travel estimates Pitch diameter and belt travel per revolution define the velocity ratio and belt length needed for a synchronous drive. The pitch line sits inside the belt, so pitch diameter is the reference dimension for timing — not pulley OD.
Important limits of the calculation
All results are theoretical. The calculator applies closed-form pitch diameter formulas. Outputs are idealized mathematical values, not measured or inspected dimensions.
This calculator does not replace design standards. Gear values do not account for AGMA or ISO quality grades, surface finish, or load ratings. Thread values are basic geometry — not tolerance class limits per ANSI/ASME B1.1, B1.13M, or ISO 724/965. Sprocket outside profile is an approximation; full tooth profile requires roller diameter, seating curve, and working curve geometry per ANSI B29.1 or ISO 606. Timing pulley OD requires belt pitch-line distance and tooth profile data not present in the pitch diameter formula.
Always verify critical parts against the applicable design standard, manufacturer catalog, or engineering drawing before specifying or manufacturing components.
References
- Spur Gear Fundamentals — Pitch Diameter, Module, Circular Pitch KHK Gear Knowledge Library. Covers spur gear pitch diameter formula PD = N × m (metric) and PD = N / DP (inch), module definition, circular pitch CP = π × m, and addendum a = m.
- Spur Gear Terminology and Formulas — Diametral Pitch, OD, Addendum, Working Depth Grob Inc. Spur Gear Terminology & Formulas. Covers diametral pitch DP = N/PD, outside diameter OD = (N+2)/DP, addendum a = 1/DP, working depth WD = 2/DP, and tooth thickness t = π/(2 × DP).
- ISO Metric Screw Thread Geometry — Basic Pitch Diameter, Minor Diameters ISO 724 and ISO 965, with d2 = d − 0.6495P, D1 = d − 1.08253P, d3 = d − 1.22687P. ASME B1.1 covers equivalent inch series definitions.
- Sprocket Pitch Diameter and Outside Diameter Formulas Tsubaki Engineering Data (U.S. Tsubaki Power Transmission LLC). Documents sprocket pitch diameter formula PD = P / sin(π/N) and the standard OD approximation for roller chain drives.
- Timing Pulley Pitch Diameter and Belt Pitch Line Pfeifer Industries LLC, Timing Belt Pulley technical data. Documents PD = (N × P)/π and notes that physical pulley OD is always smaller than theoretical PD by the belt pitch-line distance, which varies by belt profile and pitch.