Internal Ring Gear Design

Exact internal (ring) gear geometry from module or diametral pitch — teeth on the inside, cut by a pinion-type shaper cutter, measured between pins with an inside micrometer. The sheet back-solves the as-built tooth from the measurement and exports a true STEP AP203 annulus, the DXF profile, or an STL.

Updated: 7/4/2026

The live solid — drag to orbit. Every number, the exact profile and the exports follow below.

1 Type what you know — teeth + one size is enough 2 Check it between pins — or type what the shop measured 3 Read every diameter & thickness 4 Inspect the exact profile 5 Export STEP / DXF / STL

1 · The ring — size, form & blank

Metric — module English — DP

Teeth z

Diametral pitch P_d

1/in

Pressure angle α

Profile shift x

auto: valid tips

Thinning Δs

Rim OD (blank)

Min wall

Face width

Chord tolerance

Reading a print with half the numbers? Type any two independent facts below (or one plus the tooth count above) — the rest fill in gray as live results. Type over any gray value to redefine the gear; clear a field to hand it back. A typed minor Ø or major Ø is honored exactly in every view and export — the blank is bored (and the cutter fed) to the print, not to the textbook formula.

Diameters

Minor Ø (tips)

Pitch Ø

Major Ø (root)

Base Ø

Pitches & tooth

Circular pitch

Base pitch

Tooth thickness

Chordal thickness

Chordal depth

Addendum

Dedendum

Whole depth

A standard full-depth 20° ring (72 T) loads immediately, and everything re-solves the moment you stop typing. Count the teeth and mic the bore across the tips — that is enough. Entering the pitch or module directly always works too; typing it hands the helper fields back to output duty. Small rings beware: below ≈34 teeth (20°, full depth) the tips dip inside the base circle where no involute exists — the sheet refuses and names the cures (stub teeth, negative shift, more teeth).

auto: optimal shaper cutter — stock size, anti-trimming margin, max corner default

Cutter teeth z_c

Addendum h_a*

Dedendum h_f*

Cutter tip radius ρ*

Auto cutter is ON: z_c = 32 T — the nearest stock shaper to a 4-inch pitch diameter, held ≥ 12 teeth under the ring against tip trimming — with the largest corner the space accepts inside the 0.25·m practice line, ρ* = 0.250. Untick to spec your own cutter.

A hob cannot reach inside a ring — internal teeth are shaped, skived or broached, and the root fillet in every view is the true generated trochoid of this cutter's tip corner rolling on the working circles, not a pasted arc. The cutter's required tip reach (and, for thin-tooth rings, its ground shift) is derived from the ring's thickness and root, and reported in the dimensions table — resharpened or special cutters show up there honestly.

2 · Measure between pins — and set the profile from what the shop measured

Reading drives

profile shift x thinning Δs

Pin / wire Ø

Measured M (between)

→ shift x

→ thinning Δs

Pin diameter (auto — contacts at pitch Ø)

0.20734 in

Measurement BETWEEN pins (inside mic)

8.72387 in

Pin contacts flank at Ø

9.00000 in

Even tooth count — pins sit diametrically opposite. Remember the internal sign: a BIGGER pin sits deeper and reads a SMALLER dimension.

The internal-gear shop loop: shape the ring, drop two pins in opposite spaces, and read BETWEEN them with an inside micrometer. This sheet runs the loop backwards, LIVE: type what the mic read and it drives the whole model as you type — the reading stays in its field, Reading drives chooses whether it owns the profile shift x (the classic decode of an unknown ring, the default) or the thinning Δs with the print's shift trusted (bore and root stay put; only the tooth thins, exactly like feeding the shaper deeper). The owned field is locked and follows the reading; clear the reading and your own numbers come back untouched. The two decode fields beside the reading always show both interpretations — click one to copy it if you want to keep it in the design. A blank pin field bakes the suggested pin in the moment M starts driving, so the assumed pin cannot drift with the shift. Even tooth counts measure straight across; odd counts get the cos(90°/z) correction automatically. And a note for spur-gear hands: span (base tangent) measurement does not exist for internal gears — there is no outside surface to lay the anvils across. Pins are the method, and the sheet checks that the pin actually seats on the involute, clears the root, and pokes into the bore where the mic can touch it.

3 · Every dimension that matters

Diametral pitch	8.0000 1/in
Teeth z · pressure angle	72 · 20.00°
Profile shift x +x thickens the ring tooth	0.0000
Pitch diameter d	9.00000 in
Base diameter d_b	8.45723 in
Minor Ø (tooth tips — the bore)	8.75000 in
Major Ø (root)	9.31250 in
Form diameter (true SAP / TIF) involute → fillet hand-off (outward)	9.25380 in
Rim OD (blank) backup ratio m_B = 1.33	10.06250 in
Circular pitch p	0.39270 in
Base pitch p_b	0.36902 in
Tooth thickness s @ pitch	0.19635 in
Chordal thickness s̄ gear-tooth caliper	0.19633 in
Chordal depth h̄ caliper depth, set from the tip lands	0.12393 in
Tip-land thickness s_a at the minor Ø	0.11281 in
Roll angles tip / pitch / form note the reversed order on internals	15.21° / 20.85° / 25.45°
Shaper cutter in play OD 4.31250 in · α_w = 20.00°	32 T auto · reach 1.250·m · fresh grind

4 · The exact profile

Scroll to zoom, drag to pan. 72 exact internal teeth — involute flanks from the base circle, true shaper-cutter trochoid root fillets (a 32- — static preview; the interactive view loads with JavaScript.

Scroll to zoom, drag to pan. 72 exact internal teeth — involute flanks from the base circle, true shaper-cutter trochoid root fillets (a 32-tooth cutter), sampled at 0.01 thou·10⁻³ chordal tolerance — 20,160 points.

5 · Take it with you — STEP, DXF & STL

STEP AP203 is the real thing — an analytic boundary representation, not facets: planar annular caps, extruded walls, exact tip and rim circles, and cubic B-splines interpolating the certified profile points. Units are declared in the file (inch or millimetre per your toggle), so it imports at true scale. The DXF is one closed polyline of the toothed bore (plus the rim OD circle) at the chord tolerance above; the STL is the watertight faceted annulus — STL carries no units, so tell your slicer it is in inches (English mode) or millimeters (metric mode).

Parasolid (.x_t)? Parasolid is Siemens’ closed kernel format — there is no faithful third-party writer, so this sheet will not pretend to make one. The standard route: import the STEP into any Parasolid-based CAD (SolidWorks, NX, Solid Edge) and Save As → .x_t; the involute arrives as smooth analytic geometry, and the kernel does the conversion natively.

6 · The solid — rim, face width, every tooth

Watertight ring solid: 57,600 triangles, 28,800 welded vertices · volume 15.756 in³ · drag to orbit. The STL button exports exactly this mes — static preview; the interactive view loads with JavaScript.

Watertight ring solid: 57,600 triangles, 28,800 welded vertices · volume 15.756 in³ · drag to orbit. The STL button exports exactly this mesh.

I · The flank, inside out. A ring tooth's working flank is the same involute of the base circle r_b = r cos α — used on its CONCAVE side. Thickness opens OUTWARD: ψ(r) = s/d − inv α + inv φ_r, cos φ_r = r_b/r, so the tooth is thinnest at the tips (the minor Ø — the pre-machined bore) and thickest at the root (the major Ø). Because the involute only exists outside the base circle, a ring tip that dips inside it has no flank there: that is why full-depth 20° internals need roughly 34 teeth or more, and why this sheet refuses below the limit instead of drawing fiction.

II · The root, cut by a real cutter. The fillet is the envelope of the shaper cutter's tip-corner circle as the cutter (an external gear of z_c teeth) rolls INSIDE the ring on the true generation circles. Conjugacy fixes everything: the ring's thickness sets the working pressure angle through inv α_w = inv α + 2(x − x_c) tan α/(z − z_c), the base-circle identity a·cos α_w = r_b − r_b,c sets the feed depth, and the cutter's tip reach is whatever produces the specified root at that depth. The generated point obeys the law of gearing — its normal passes through the rolling pitch point — so the envelope is explicit, the root tangency is closed-form, and the fillet–involute hand-off happens at the roll instant where the contact crosses the cutter's own flank-to-corner blend. Change the cutter tooth count and the fillet honestly changes with it.

III · Between pins. Two pins in opposite spaces, an inside mic between them. The closure mirrors the external case with the pin on the concave flank: inv φ_M = inv α + π/z − s/d − d_p/d_b, center circle r_M = r_b/cos φ_M, M = 2 r_M − d_p (even z; odd z gets the cos 90°/z correction). Note the internal signs: a BIGGER pin sits DEEPER and reads SMALLER. Run backwards — measured M → φ_M → thickness s → shift — it recovers the as-cut ring, which is exactly how this sheet adjusts the profile. The engine cross-checks the closed form against a brute-force geometric pin-contact solve to ~10⁻⁷, flags pins that seat in the fillet, ride the tips, bottom in the root, or hide outside the bore where no mic can touch them.

IV · Accuracy. All geometry is closed-form in double precision; curves are emitted as full-precision polylines refined below your chordal tolerance (default m/10,000). The engine's ring suite verifies the involute by its defining property (every flank normal is tangent to the base circle — the string test, applied point-by-point to the emitted profile), proves the between-pins closed form against an independent geometric contact solve, checks the zero-backlash duality against a standard external pinion (0.25·m clearances at both ends), and confirms the annulus solid watertight with the volume matching the cap-area × width identity to 10⁻⁶. The STEP export is re-parsed and its edge graph checked closed (every edge used exactly twice, opposite senses).

Notes on using the results

The minor Ø is a bore, the major Ø is the cut. The blank is bored to the minor diameter first — the tooth tips are remnants of that surface — and the shaper cuts the spaces outward to the major diameter. That is why thinning Δs moves the major Ø and every measurement while the minor Ø stays put, and why this sheet honors a typed minor/major exactly.
Backup ratio. The rim behind the root carries the mesh: rim wall / whole depth below ≈1.2 and the ring breathes under load and distorts in the chuck. The sheet defaults the blank OD to a ratio of ≈1.33 and warns when you go thin.
Shaper-cut roots are not rack-table roots. On shifted or thinned rings the cutter reach required to hit the rack-formula major Ø drifts from the standard 1.25·m — the dimensions table reports the reach in play, and extreme demands are flagged (or refused when the corner would trim the flank away). If a print's root disagrees with the formula, type it: the trochoid follows the print.
Backlash & tolerances. The Thinning Δs field carries the allowance separately from the design shift, and the between-pins measurement can be attributed to either (x for an unknown ring, Δs for a known print).
Mating checks live on the pair. Involute interference with the pinion, trochoid interference, and the radial-assembly limit (a pinion can only be dropped in radially when z₂ − z₁ is large enough — otherwise it goes in axially) are properties of the PAIR, not this single ring, and belong to the mating sheet planned next. The cutter-side equivalents ARE checked here, because the cutter is part of this ring's definition.

References & further reading

tec-science — Profile shift — the x = 1 − z·sin²α/2 no-undercut relation this sheet applies in its cutter-true form.
Machinery's Handbook — Gearing section: internal gears, measurement between pins/wires, profile-shift practice.
KHK — Gear Technical Reference — internal gear geometry, interference limits, between-pin tables.
Colbourne, The Geometry of Involute Gears — the signed-tooth-number convention and internal gearing theory.
Litvin & Fuentes, Gear Geometry and Applied Theory — generation, envelopes and shaper-cut fillets.
Wikipedia — Gear shaper — how internal teeth are actually cut.

Disclaimer

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