Spur Gear Module & Diametral Pitch Calculator: Pitch Dia, Blank OD & Span W_k

How Do Metric Gear Module (m) and Imperial Diametral Pitch (DP) Relate Mathematically?

Every involute spur gear mesh requires two mating gears to share the exact same tooth spacing along their rolling pitch circles—called the Circular Pitch (p)—and the same Pressure Angle (α), standardized at 20° under ISO International Metrology Standard (ISO 53) and ASME Geometric & Mechanical Standards / AGMA 2001-D04. Because circular pitch contains the irrational number π (p = π × D / z), dimensioning gear drawings directly by circular pitch would produce awkward irrational center distances. To make pitch diameters and shaft center distances clean rational numbers, engineers divide out π in one of two reciprocal ways.

In the ISO metric system (ISO 53 / DIN 867), the tooth size index is the Module (m), defined in millimeters as the pitch diameter divided by the number of teeth: m = D / z = p / π. A larger Module means a larger, stronger gear tooth. Conversely, in the American AGMA inch system, the tooth size index is the Diametral Pitch (DP or P_d), defined as the number of teeth per inch of pitch diameter: DP = z / D_in = π / p_in. Because m is millimeters per tooth and DP is teeth per inch, converting between the two systems is an exact inverse relation across 25.4 mm/in: m = 25.4 / DP and DP = 25.4 / m.

How Do You Calculate Blank Outside Diameter, Root Diameter, and Mating Center Distance?

Before mounting a gear blank on a hobbing machine, rotary 4th-axis CNC mill, or gear shaper, the lathe operator must turn the outer cylindrical blank diameter (D_o) to a tight tolerance (typically ISO h7 or h8) because gear cutters are often touched off directly on the outside rim. For a standard full-depth 20° involute gear without profile shift (x = 0), the Addendum (a)—the radial height of the tooth above the pitch circle—equals 1.00 × m (1.00 / DP). Adding two addendums to the pitch diameter gives the universal blank turning equation: D_o = D + 2a = m × (z + 2) = (z + 2) / DP.

To prevent the tip of the mating gear tooth from rubbing at the bottom of the tooth space, the Dedendum (b)—the radial depth below the pitch circle—includes a 0.25 × m root clearance (or 0.157 × m on legacy Brown & Sharpe cutters): b = 1.25 × m, yielding a total cutting Whole Depth of h = a + b = 2.25 × m (or 2.157 / DP) and a Root Diameter of D_r = m × (z - 2.5). When two external spur gears with tooth counts z_1 and z_2 mesh together, their nominal shaft-to-shaft Center Distance is exactly half the sum of their pitch diameters: C = (D_1 + D_2) / 2 = m × (z_1 + z_2) / 2.

How Do Toolmakers Inspect Gear Tooth Thickness Using Base Tangent Span Over k Teeth (W_k)?

Measuring individual tooth chordal thickness with a gear-tooth vernier caliper relies on the outside diameter D_o as a reference—meaning any runout or burrs on the turned blank OD corrupts the reading. Instead, precision gear shops inspect tooth thickness using the Wildhaber Base Tangent Span Method (W_k) traceable to Machinery's Handbook (Industrial Press) and NIST (National Institute of Standards and Technology) involute geometry. By placing the flat parallel anvils of a disc (flange) micrometer across k consecutive teeth, the anvils contact the opposing involute profiles along a tangent line to the Base Circle (D_b = D × cos α), making the measurement 100% independent of blank OD errors!

Using the involute function inv(α) = tan(α) - α_rad (where inv 20° = 0.01490438 and cos 20° = 0.9396926), the optimal number of teeth to span is k = round[(z × α_deg / 180) + 0.5], and the exact theoretical zero-backlash span measurement is W_k = m × cos(α) × [(k - 0.5) × π + z × inv(α)]. After hobbing your gear, size the shaft keyway on our ANSI B17.1 & DIN 6885 Keyway Calculator or ANSI B17.2 Woodruff Keyseat Calculator, check hub press fits on our Shaft & Hub Press-Fit Calculator, and verify mounting hole patterns on our Machinist Bolt Circle Calculator (PCD).