Pitch diameter shows up in two completely different machine-shop contexts that happen to share a name: the fit-determining diameter of a screw thread, and the reference rolling circle of a gear. This calculator handles both, because most searches for "pitch diameter" could mean either one depending on whether the reader is inspecting a fastener or designing a gear train.
Why thread pitch diameter matters more than major diameter
A bolt's major diameter is the number stamped on the part — 1/4 in, M10, and so on — but it is not the dimension that determines whether the bolt actually threads into a nut with the correct amount of engagement. That job belongs to the pitch diameter, the diameter of the imaginary cylinder where the thread groove and the thread land are exactly equal in width. Thread gauges (go/no-go ring and plug gauges) check pitch diameter directly, and thread fit classes (2A/2B for general purpose, 3A/3B for tight tolerance) are specified as a pitch-diameter tolerance band around the theoretical value this calculator computes.
The formula PD = Major Diameter − 0.6495 × Pitch is the standard approximation for external 60-degree V-threads used by both the Unified inch system (UNC, UNF, UNEF) and ISO metric threads. The constant comes from the geometry of a 60-degree thread form: the theoretical sharp-V thread height is 0.866025 × pitch, and standard threads truncate both the crest and root, leaving 0.6495 × pitch as the depth from major diameter down to the pitch line.
Gear pitch diameter and the two sizing systems
A gear's pitch diameter is the diameter of the imaginary friction-disc that would produce the same rotational motion as the actual meshing teeth. Every other gear dimension — addendum, dedendum, whole depth, center distance between two meshing gears — is derived from this one reference circle, which is why it is the first number to nail down when specifying or reverse-engineering a gear.
Inch-system gears describe tooth size with diametral pitch (DP): teeth per inch of pitch diameter, so PD = teeth / DP. Metric-system gears use module instead: millimeters of pitch diameter per tooth, so PD = teeth × module. The two systems are reciprocal (module = 25.4 / DP) but not directly interchangeable — a 10 DP gear (module ≈ 2.54) will not mesh cleanly with a true module-2.5 gear even though the tooth sizes are close, because the tooth profiles are cut to slightly different standards.
Limits and edge cases
The thread formula assumes a standard external 60-degree V-thread profile (UN/UNC/UNF/ISO metric) — it does not apply to Acme, buttress, pipe (NPT), or other non-60-degree thread forms, which use different depth constants. It also computes the theoretical basic pitch diameter, not a specific tolerance-class value; real parts are cut slightly undersize (external) or oversize (internal) within a fit-class tolerance band around this number.
The gear formula computes the standard full-depth pitch diameter for spur gears with a given tooth count and diametral pitch or module. It does not account for profile shifting (addendum modification), helical gear normal-vs-transverse pitch differences, or internal (ring) gears, all of which need additional inputs beyond teeth and DP/module. For any load-bearing or precision-fit application, verify against the applicable ANSI/AGMA or ISO gear standard and, for threads, against ASME B1.1 (Unified) or ISO 68-1 (metric).