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Thread Dimensions Calculator

Pitch, minor and major diameters for any 60 degree inch or metric thread, with three wire measurement, best wire size and tensile stress area.

How to use this calculator

  1. 1Pick a standard thread, or enter any diameter and pitch - the geometry works for combinations no table lists.
  2. 2For single point threading on a lathe, cut to the over-wires reading rather than to a diameter; it is the only practical way to hit the pitch diameter.
  3. 3Enter a specific pitch diameter if you are working to a class limit rather than to basic.
  4. 4Use wires within about 10% of the best size. Outside that the reading becomes sensitive to thread angle errors.

How the calculation works

H = sqrt(3)/2 x pitch (the sharp V height) Pitch diameter = major - 0.649519 x P (= 2 x 3H/8) Minor, internal = major - 1.082532 x P (= 2 x 5H/8) Minor, external = major - 1.226869 x P (= 2 x 17H/24) Best wire = 0.57735 x P Over wires M = E + 3W - 0.86603 x P
H
The height of the sharp 60 degree triangle the thread is truncated from. Everything else is a fraction of it
E
Pitch diameter - the diameter at which thread and groove are equal width. It is what governs fit, and what a thread gauge checks
W
Wire diameter. The best size touches the flank exactly at the pitch line, which cancels thread angle errors
0.86603
cot(30)/2 x ... in the compact form. Using the major diameter instead, the same formula is M = D - 1.51554 P + 3W

Unified and ISO metric threads share the same 60 degree form and the same fractions of H. Only the way the pitch is specified differs - threads per inch against millimetres.

The external thread's minor diameter is larger than the internal one because its root is rounded to 17H/24 rather than cut flat to 5H/8. That clearance is deliberate: it is where the stress concentration would otherwise be.

These are basic dimensions. A class of fit applies allowances and tolerances on top, and for a 2A external thread the pitch diameter maximum sits below basic by a small allowance.

Worked example

A 1/2-13 UNC, the classic

  1. 1.Pitch is 1/13 = 0.076923 in, so H = 0.866025 x 0.076923 = 0.066617 in.
  2. 2.Pitch diameter = 0.5 - 0.649519 x 0.076923 = 0.4500 in.
  3. 3.Internal minor = 0.5 - 1.082532 x 0.076923 = 0.4167 in; external minor = 0.5 - 1.226869 x 0.076923 = 0.4056 in.
  4. 4.Best wire is 0.57735 x 0.076923 = 0.04441 in, and over three of them the reading is 0.4500 + 3(0.04441) - 0.86603(0.076923) = 0.51665 in.

Result: 0.4500 pitch diameter, 0.51665 over wires

M12 x 1.75, the metric equivalent

  1. 1.H = 0.866025 x 1.75 = 1.5155 mm - the same relation, just stated in millimetres.
  2. 2.Pitch diameter = 12 - 0.649519 x 1.75 = 10.863 mm.
  3. 3.Internal minor = 12 - 1.082532 x 1.75 = 10.106 mm, which is the honest 100% tap drill for M12.
  4. 4.Best wire is 1.0104 mm and the over-wires reading is 12.3789 mm.

Result: 10.863 pitch diameter, 12.379 over wires

Cutting to a class limit on a lathe

  1. 1.A 2A external 1/2-13 has a pitch diameter around 0.4435 to 0.4485 - below basic by the allowance and tolerance.
  2. 2.Aiming at 0.4435, the over-wires reading becomes 0.4435 + 0.133236 - 0.066617 = 0.51012 in.
  3. 3.So the lathe is fed in until the wires read 0.5101, and the pitch diameter is right whatever the crest looks like.
  4. 4.This is the only practical way to hit a pitch diameter on a single point thread - there is nothing to measure directly.

Result: Cut until the wires read 0.5101

An unusual thread no table lists

  1. 1.A 13/16-18 is not a standard combination, but the geometry does not care.
  2. 2.Pitch 1/18 = 0.055556, so pitch diameter = 0.8125 - 0.649519 x 0.055556 = 0.77641 in.
  3. 3.Internal minor = 0.8125 - 1.082532 x 0.055556 = 0.75236 in.
  4. 4.Best wire 0.032075, over wires 0.82453. Nothing about the calculation needed the thread to be standard.

Result: 0.77641 pitch diameter on a thread no table covers

One triangle, three fractions

Both Unified and ISO metric threads are the same 60 degree form. Start with a sharp V of that angle at the given pitch: its height H is the pitch times the sine of 60 degrees, which is sqrt(3)/2, or 0.866025.

A sharp V cannot be manufactured or used - the crest would be a knife edge and the root an infinitely sharp notch. So the standard truncates both. The crest is cut back H/8 and the root H/4, and everything follows from those two decisions.

The pitch diameter, where thread and groove are equal width, sits 3H/8 in from the major diameter on each side, so 0.649519 times the pitch on diameter. The internal thread's minor is 5H/8 per side, or 1.082532 P. The external thread's root is rounded and sits at 17H/24, or 1.226869 P. Those three constants are the entire table, and they work for any diameter and pitch anyone ever specifies.

Why pitch diameter is the one that matters

A screw and a nut do not touch at their crests or their roots. Both are deliberately clearanced there, because that is where manufacturing variation and stress concentration live. They touch on the flanks, and the flanks are located by the pitch diameter.

This makes the pitch diameter the dimension that determines whether a thread fits, whether it is loose, and how much of the load each turn carries. Classes of fit are almost entirely tolerances on it. A go/no-go ring gauge is checking it. A thread that measures perfectly on its major diameter and will not accept a nut has a pitch diameter problem.

It is also the dimension you cannot measure directly. There is no surface at the pitch diameter to put a micrometer on - it is an imaginary cylinder inside the thread form. Hence the wires.

How three wires reach a diameter that is not there

Lay three wires of equal size in the thread grooves - two adjacent on one side, one opposite between them, so a micrometer sits square - and measure across. The wires rest on the flanks, and if their diameter is chosen correctly they touch exactly at the pitch line.

The geometry then relates the reading to the pitch diameter directly: M = E + W(1 + cosec 30) - (P/2) cot 30, which for a 60 degree thread simplifies to M = E + 3W - 0.86603 P. Written from the major diameter instead, the same relation is M = D - 1.51554 P + 3W, which is the form most machinist references print.

The "best" wire size, 0.57735 times the pitch, is the one that contacts precisely at the pitch line. That matters for a subtle reason: at exactly that contact point, errors in the thread angle move the two flanks in opposite directions and cancel. Away from the best size they do not, and the measurement starts reporting angle error as pitch diameter error. Wires within about 10% of best are fine; well outside, the reading needs treating with suspicion.

For anyone single point threading on a lathe this is the practical technique. Cut, measure over wires, and feed in until the reading matches - the pitch diameter is then right regardless of what the crests look like.

Tensile stress area, and why it is neither diameter

Ask what area a bolt breaks across and the obvious answers are wrong. It is not the major diameter area - that includes thread crests that carry nothing. It is not the minor diameter area either, which is the answer most people guess.

The reason is the helix. A thread is a continuous spiral, so no plane cross section of the bolt is entirely at the root; each one cuts through the thread at a different depth around the circumference. The effective area comes out between the pitch and minor diameters, and the standards define it empirically as pi/4 times (D - 0.9743/n) squared in inch units, or (D - 0.9382 P) squared in metric.

That is the number every published fastener strength is based on. A 1/2-13 has a stress area of 0.1419 square inches, so a grade 8 bolt at 150,000 psi proof holds about 17,000 pounds. Using the minor diameter area instead would understate it by around 15%, and using the major would overstate it by about the same.

What this assumes, and where it stops

Assumptions

  • Basic dimensions per ASME B1.1 and ISO 68-1. Classes of fit apply allowances and tolerances on top of these.
  • A 60 degree thread form with the standard H/8 crest and H/4 root truncations.
  • Single start threads. Multi-start threads have a lead that is a multiple of the pitch, which changes the helix angle but not the profile.
  • The over-wires formula ignores the lead angle correction, which is small below about 4 degrees of helix.
  • Tensile stress area uses the standard empirical formula rather than an exact integration.

Limitations

  • It gives basic dimensions, not class limits. For a fit class you need the allowance and tolerance from the standard as well.
  • It covers 60 degree threads only. Acme, buttress, Whitworth at 55 degrees and pipe threads all use different constants.
  • The lead angle correction to the wire measurement is not applied - significant on coarse pitches, fine threads on small diameters, and multi-start work.
  • It does not calculate thread strength, engagement length or torque.
  • Measurement over wires assumes wires resting on the flanks. Very small wires bottom in the root and give a meaningless reading.

Common questions

How do I calculate thread pitch diameter?

Subtract 0.649519 times the pitch from the major diameter. For a 1/2-13 the pitch is 1/13 = 0.076923, so the pitch diameter is 0.5 - 0.049963 = 0.4500 in. The same constant works for metric with the pitch in millimetres: M12 x 1.75 gives 12 - 1.1367 = 10.863 mm.

What is the best wire size for measuring a thread?

0.57735 times the pitch, which is 1/sqrt(3). At that diameter the wire touches the thread flank exactly at the pitch line, so errors in the thread angle cancel rather than showing up as pitch diameter error. Anything within about 10% of it is usable.

What is the three wire measurement formula?

M = E + 3W - 0.86603 x P, where E is the pitch diameter, W the wire diameter and P the pitch. Written from the major diameter it is M = D - 1.51554 x P + 3W. For a 1/2-13 at basic pitch diameter with best wires, that is 0.51665 in.

Is the minor diameter the same for the screw and the nut?

No. The internal thread's minor diameter is major minus 1.082532 times the pitch; the external thread's is major minus 1.226869 times the pitch, because its root is rounded to a larger truncation. The external one is smaller, and the difference is deliberate clearance at the root where the stress concentration would otherwise sit.

What is tensile stress area and why is it not the minor diameter?

It is the effective section a bolt breaks across, and it falls between the pitch and minor diameters. Because a thread is a helix, no single cross section is entirely at the root - each cuts the thread at a different depth around the circumference. The standard formula is pi/4 times (D - 0.9743/n) squared, and every published fastener strength is based on it.

Sources

Formula and content last reviewed on .

Results are estimates for information only, not professional advice.

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