ISO Metric Trapezoidal Thread Calculator (DIN 103 Tr): Dimensions & 3-Wire Chart
Under ISO 2901 and DIN 103, an ISO Metric Trapezoidal power screw thread (designated Tr D × P, such as Tr 20×4) has a symmetric 30° included flank angle (15° half-angle), a Basic Pitch Diameter of d_2 = D - 0.500 × P, an internal nut Tap Drill / Minor Diameter of D_1 = D - P, and a Best Wire Size of w_best = 0.51764 × P.
ISO Metric 30° Trapezoidal Thread Calculator (DIN 103 Tr Series)
Compute Pitch Diameter (d_2), External Root (d_3), Internal Tap Drill (D_1), Class 7e/7H limits, and 3-Wire MOW (M_w).
How Does an ISO Metric 30° Trapezoidal Thread (Tr) Differ from an American 29° Acme Thread?
Throughout European and Asian machine tool manufacturing, 3D printers, CNC routers, scissor lifts, and linear actuators, bidirectional leadscrews are standardized under ISO International Metrology Standard (ISO 2901 / ISO 2902 / ISO 2903) and German standard DIN 103 as Metric Trapezoidal Threads (Tr). While American Acme threads governed by ASME Geometric & Mechanical Standards (ASME B1.5) use an imperial 29° included groove angle (14.5° half-angle) and Threads Per Inch (TPI), ISO Metric Trapezoidal threads use a clean 30° included groove angle (15.0° half-angle) and millimeter pitch (P).
From the 30° fundamental triangle, the theoretical sharp-V height is H_1 = P / [2 × tan(15°)] = 1.866025 × P, and the basic engagement height is exactly half the pitch: H_2 = 0.5000 × P. To prevent top-land rubbing and provide a grease reservoir at the root of the thread, DIN 103 adds a radial crest/root clearance a_c (0.15 mm for P ≤ 1.5 mm, 0.25 mm for P = 2 to 5 mm, and 0.50 mm for P = 6 to 12 mm). This yields a total external tooth height of h_3 = 0.500 × P + a_c, an external screw minor diameter of d_3 = D - 2·h_3 = D - P - 2·a_c, and an internal nut major diameter of D_4 = D + 2·a_c.
| Designation (Tr D × P) | Pitch P & Crest Clearance a_c | Basic Pitch Dia d_2 = D - 0.5P | Screw Minor Dia d_3 = D - P - 2a_c | Nut Minor / Tap Drill D_1 = D - P | Nut Major Dia D_4 = D + 2a_c | Best Wire w = 0.5176P |
|---|---|---|---|---|---|---|
| Tr 10 × 2 | P = 2.0 mm (a_c = 0.25) | 9.000 mm (0.3543") | 7.500 mm (0.2953") | 8.000 mm (0.3150") | 10.500 mm | 1.035 mm |
| Tr 12 × 3 | P = 3.0 mm (a_c = 0.25) | 10.500 mm (0.4134") | 8.500 mm (0.3346") | 9.000 mm (0.3543") | 12.500 mm | 1.553 mm |
| Tr 16 × 4 | P = 4.0 mm (a_c = 0.25) | 14.000 mm (0.5512") | 11.500 mm (0.4528") | 12.000 mm (0.4724") | 16.500 mm | 2.071 mm |
| Tr 20 × 4 | P = 4.0 mm (a_c = 0.25) | 18.000 mm (0.7087") | 15.500 mm (0.6102") | 16.000 mm (0.6299") | 20.500 mm | 2.071 mm |
| Tr 24 × 5 | P = 5.0 mm (a_c = 0.25) | 21.500 mm (0.8465") | 18.500 mm (0.7283") | 19.000 mm (0.7480") | 24.500 mm | 2.588 mm |
| Tr 30 × 6 | P = 6.0 mm (a_c = 0.50) | 27.000 mm (1.0630") | 23.000 mm (0.9055") | 24.000 mm (0.9449") | 31.000 mm | 3.106 mm |
| Tr 40 × 7 | P = 7.0 mm (a_c = 0.50) | 36.500 mm (1.4370") | 32.000 mm (1.2598") | 33.000 mm (1.2992") | 41.000 mm | 3.623 mm |
| Tr 50 × 8 | P = 8.0 mm (a_c = 0.50) | 46.000 mm (1.8110") | 41.000 mm (1.6142") | 42.000 mm (1.6535") | 51.000 mm | 4.141 mm |
How to Inspect DIN 103 Class 7e Leadscrews Over 3 Wires on a CNC Lathe
Under ISO 2903 and Machinery's Handbook (Industrial Press), general-purpose precision leadscrews pair an external steel screw machined to tolerance class 7e (which incorporates a negative fundamental allowance es for smooth running lubrication film) with a cast bronze or Ertalyte nut tapped or bored to tolerance class 7H (zero fundamental deviation, EI = 0). Because standard 60° pitch micrometers cannot enter a 30° trapezoidal groove, machinists measure the external pitch diameter d_2 using the Three-Wire Method traceable to NIST (National Institute of Standards and Technology) metrology formulas.
For a symmetric 30° groove (α / 2 = 15°), the wire diameter that contacts the flanks at the exact pitch line is w_best = P / [2 × cos(15°)] = 0.517638 × P. Because cot(15°) = 3.732051 and 1 + 1 / sin(15°) = 4.863703, the exact micrometer Measurement Over 3 Wires (M_w) for any ISO 30° Trapezoidal thread is:
M_w = d_2 - 1.866025 × P + 4.863703 × w (With Best Wire: M_w = d_2 + 0.65161 × P)
| Thread Size | Screw 7e Allowance (es) | Screw 7e Pitch Dia Limits (d_2) | Nut 7H Pitch Dia Limits (D_2) | 3-Wire Reading M_w with Best Wire (7e Screw) |
|---|---|---|---|---|
| Tr 12 × 3 | -0.085 mm | 10.165 to 10.415 mm (0.4002"–0.4100") | 10.500 to 10.835 mm | 9.226 to 9.476 mm (w = 1.553 mm) |
| Tr 16 × 4 | -0.095 mm | 13.605 to 13.905 mm (0.5356"–0.5474") | 14.000 to 14.400 mm | 12.353 to 12.653 mm (w = 2.071 mm) |
| Tr 20 × 4 | -0.095 mm | 17.605 to 17.905 mm (0.6931"–0.7049") | 18.000 to 18.400 mm | 16.353 to 16.653 mm (w = 2.071 mm) |
| Tr 24 × 5 | -0.106 mm | 21.059 to 21.394 mm (0.8291"–0.8423") | 21.500 to 21.950 mm | 19.493 to 19.828 mm (w = 2.588 mm) |
| Tr 30 × 6 | -0.118 mm | 26.507 to 26.882 mm (1.0436"–1.0583") | 27.000 to 27.500 mm | 24.629 to 25.004 mm (w = 3.106 mm) |
| Tr 40 × 7 | -0.125 mm | 35.975 to 36.375 mm (1.4163"–1.4321") | 36.500 to 37.030 mm | 33.782 to 34.182 mm (w = 3.623 mm) |
Multi-Start Trapezoidal Leadscrews (Lead L vs Pitch P) and Self-Locking Efficiency
When a rapid linear actuator specifies a multi-start designation such as Tr 20×8 (P4), the first number after the cross (8 mm) is the axial Lead (L)—the linear distance the nut travels in one revolution—while P4 specifies the axial Pitch (P = 4 mm) between adjacent teeth (indicating a 2-start thread: Starts = L / P = 8 / 4 = 2). Always size your threading insert and 3-wire inspection to the Pitch P, but program your CNC lathe feed rate to the Lead L! Compare 29° inch screws on our 29° Acme & Stub Acme Thread Calculator, asymmetric high-thrust screws on our 7°/45° Buttress Thread Calculator, 60° wire readings on our Three-Wire Thread Measurement Calculator, and G-code passes on our Fanuc G76 Threading Cycle Calculator.