§1The buried diameter
Major diameter is a calliper job and minor nearly so — but the thread’s functional size is the pitch diameter, and its cylinder passes through mid-air between the flanks.
The calculating page defined d2 as the diameter where ridge and groove are equal; the systems page located every fit and every flank bearing on it; the tolerance classes of the metric and Unified pages are, at bottom, tolerances on d2. Yet nothing physical exists at that diameter to touch: a flat anvil lands on the crests (reading the wrong, and least functional, dimension), and a knife anvil in the groove bottoms in the root radius. Every method on this page is therefore a scheme for making solid contact on the flanks — the surfaces that actually work — and converting that contact arithmetically back to d2. Three schemes cover practice: reference forms that embody the whole thread and simply fit or don’t (§2), cylinders of known size seated on the flanks and measured over (§3–4), and anvils ground to thread form that read d2 directly (§5). Behind all three stands the section’s quiet dependency: someone must measure the gauges and the anvils themselves, and §6 ends at the ground master threads where that regress stops.
Contents§2Limit gauges and Taylor’s principle
For the shop floor, the pitch diameter is not measured at all — it is bracketed, by a GO gauge that must assemble and a NOT-GO gauge that must not.
A thread plug gauge (for holes) or ring gauge (for screws) is a hardened, ground reference thread at the limit of tolerance, and the test is binary: the GO member screws fully home by fingers, the NOT-GO refuses after at most a couple of turns, and a part passing both is in tolerance — no number ever read. The design of the pair follows Taylor’s principle, the same logic the dimensioning pages applied to plain limit gauges. The GO gauge is full-form and full-length: it checks everything at once — pitch diameter, form, pitch and lead accumulated over the whole engagement — because that is exactly what assembly with the worst-case mating part will demand; anything that stops the GO gauge would stop a nut. The NOT-GO gauge checks one element only: its profile is truncated to bear on the flanks alone and it is only a few threads long, so that pitch and form errors cannot hold it out of a hole whose pitch diameter has actually strayed oversize. One gauge asks “will it assemble?”, the other “has the material limit been breached?” — and between them they define acceptance so completely that §6 will conclude the numerical methods exist largely to serve, set and settle arguments with this pair.
Contents§3Three wires: the geometry
Seat three calibrated cylinders in the thread grooves — two one side, one the other — and a plain micrometer over them reads a number that pure geometry converts to the pitch diameter.
The derivation is two right triangles. A wire of diameter G resting in a symmetric vee of half-angle α/2 sits with its centre a distance G/(2 sin(α/2)) above the vee’s apex — the deeper geometry of every wire, ball and cone location in metrology — and the apex of the thread groove sits a computable distance below the pitch line, fixed by p and the cotangent of the same half-angle. Stack the two, double for both sides, and the measurement over wires M relates to d2 by the boxed line: for 60° threads the wire factor collapses to exactly 3 and the pitch term to the familiar 0.86603 p. The formula holds for any wire slim enough to touch the flanks and fat enough to stand proud — but one size is special: the best wire, G = p/(2 cos(α/2)) = 0.57735 p for 60°, touches the flanks precisely on the pitch line, so the small errors this planar derivation ignores — the helix angle tilting the contact, any error in the thread’s own flank angle — enter at their minimum. Best-size wires come boxed in sets per pitch; the working habit is to use them, and the formula forgives you when you cannot.
Contents§4The M10, worked and tabulated
Run the standing example through the formula both ways — best wires and ordinary ones — and tabulate the constants for the section’s two vee angles.
| Quantity | 60° metric / Unified | 55° Whitworth |
|---|---|---|
| Best wire G | 0.57735 p → 0.866 mm | 0.5637 p |
| Wire factor 1 + 1/sin(α/2) | 3.0000 | 3.1657 |
| Pitch term (p/2) cot(α/2) | 0.86603 p | 0.9605 p |
| M10×1.5 over best wires | 10.325 mm | — |
| Same thread, Ø1.0 mm wires | 10.727 mm | — |
| The worked line: M = 9.026 + 3 × 0.86603 − 1.299 = 10.325 mm — measure that over best wires and the thread’s pitch diameter is at basic size. Swap in ordinary 1.0 mm drill blanks and the same formula predicts 10.727 mm: the method is not hostage to a wire set, only to knowing G. Run either backwards on a real screw — measure M, solve for d2 — and compare against the 6g band: three wires, one micrometer and four multiplications have just performed the inspection the tolerance pages promised. | ||
§5Direct readers
The three-wire method’s arithmetic can be built into the instrument: grind the anvils to thread form and the micrometer reads pitch diameter directly.
A thread micrometer carries a 60° cone on one anvil and a matching vee on the other; seated on opposite flanks of the screw, the pair contacts at the pitch line and the barrel reads d2 with no formula at all. The convenience costs two disciplines. The anvils suit a range of pitches only — the cone must reach the flanks without bottoming in fine threads or riding the crests of coarse ones — so the instrument comes with interchangeable anvil sets, chosen per job. And it must be set to a master: zeroing on a reference thread or setting standard of known d2, because the anvil geometry folds its own errors into every reading in a way the honest, dumb wires of §3 cannot. In production the same idea goes gauge-shaped: thread comparators and roll snap gauges present form-ground rolls or segments to the work and show a dial deviation from a master-set zero — no absolute number, just pass, fail and drift, read in a second at the machine. The family line is worth seeing plainly: wires are geometry you compute, thread mics are geometry built into anvils, comparators are geometry built into a fixture — one method, at three levels of throughput.
Contents§6Form, pitch and the virtual thread
Pitch diameter is not the whole thread: flank angle and pitch carry their own errors, and — decisively — those errors consume pitch-diameter tolerance as if they were size.
The optical projector throws the thread’s magnified silhouette onto a screen chart of the perfect form, and flank angle, root radius, crest condition and pitch are checked against drawn limits — the instrument of choice for form disputes, tap and gauge inspection, and §3’s ignored flank-angle term made visible. Pitch and lead are checked from gauge to instrument grade: the humble pitch gauge identifies; measuring machines track a stylus flank-to-flank and integrate lead error along the length. The decisive idea tying them to §2 is the virtual (functional) thread: a screw with perfect d2 but a lead error acts, to its nut, larger — the drunken helix must be enclosed by a bigger perfect one to assemble — so every micrometre of accumulated lead or form error is a micrometre of d2 tolerance spent. That is why the full-form GO gauge of §2 keeps the final word over any single-number measurement, why precision threads specify lead over length and not just class, and why the instruments of this page ultimately calibrate against ground master threads — which is precisely where the thread-grinding page, two ahead, takes up the story.
Contents§7Quick reference
The working core of the page on one card rack.
Target
d2 — the flank cylinder
crests tell you least
Gauging
GO full-form, full-length
NOT-GO one element, short
Three wires
M = d2 + 3G − 0.866 p
best wire G = 0.577 p
M10×1.5
best wires: M = 10.325 mm
Ø1.0 wires: 10.727 mm
Final word
lead error eats d2 tolerance
the GO gauge judges