Lathe Taper Per Foot (TPF) to Degrees Calculator: Angle & Tailstock Offset
For a conical shaft with large diameter D, small diameter d, and tapered length L_t, the Taper Per Inch is TPI = (D - d) / L_t, the Taper Per Foot is TPF = 12 × (D - d) / L_t, and the lathe Compound Rest Half-Angle is α = arctan[(D - d) / (2 × L_t)] = arctan(TPF / 24).
Lathe Taper Per Foot (TPF), Taper Angle & Tailstock Offset Calculator
Calculate Taper Per Foot (TPF), Taper Per Inch (TPI), Included Angle (2α), Compound Rest Half-Angle (α), and Tailstock Offset.
How Do You Convert Taper Per Foot (TPF) to Included Angle (2α) and Compound Rest Angle (α)?
On machine shop blueprints governed by ASME Geometric & Mechanical Standards (ASME B5.10) and ISO International Metrology Standard (ISO 1119), conical arbors, spindle noses, and shafts are specified by Taper Per Foot (TPF), Taper Ratio (1 : X), or Included Angle (2α). The Included Angle (2α) spans opposing walls, while the Half-Angle (α) measures one wall to the spindle centerline.
When swiveling a manual engine lathe's compound rest to cut a taper—or when programming a CNC lathe G01 linear interpolation vector—the cutting tool moves along a single side of the part diameter. Therefore, you must always set the lathe compound slide to the Half-Angle (α), never the Included Angle (2α)! Because radial change per unit length is half the diametral change (ΔR = (D - d) / 2), the exact trigonometric conversion from Taper Per Foot to degrees is:
Using the fundamental radial tangent equation tan(α) = (D - d) / (2 × L_t) = TPI / 2 = TPF / 24, the compound rest half-angle is α = arctan(TPF / 24) and the full included cone angle is 2α = 2 × arctan(TPF / 24). Before locking the compound slide swivel nuts, always sweep a master conical taper plug gage or precision ground test bar with a 0.001 mm (0.00005") dial test indicator mounted on the carriage saddle across the full travel to confirm zero angular runout.
| Machine Taper Standard | Taper Per Foot (TPF, in/ft) | Taper Per Inch (TPI, in/in) | Ratio (1 : X) | Included Taper Angle (2α) | Compound Rest Half-Angle (α) |
|---|---|---|---|---|---|
| Jarno Taper (All Sizes #2–#20) | 0.6000 in/ft | 0.05000 in/in | 1 : 20.000 | 2° 51' 51" (2.8642°) | 1° 25' 56" (1.4321°) |
| Brown & Sharpe (#1 to #18, exc. #10) | 0.5020 in/ft | 0.04183 in/in | 1 : 23.905 | 2° 23' 48" (2.3966°) | 1° 11' 54" (1.1983°) |
| Brown & Sharpe #10 | 0.5161 in/ft | 0.04301 in/in | 1 : 23.251 | 2° 27' 50" (2.4639°) | 1° 13' 55" (1.2319°) |
| Morse Taper MT0 | 0.6246 in/ft | 0.05205 in/in | 1 : 19.212 | 2° 58' 54" (2.9817°) | 1° 29' 27" (1.4908°) |
| Morse Taper MT1 | 0.5986 in/ft | 0.04988 in/in | 1 : 20.047 | 2° 51' 26" (2.8572°) | 1° 25' 43" (1.4286°) |
| Morse Taper MT2 | 0.5994 in/ft | 0.04995 in/in | 1 : 20.020 | 2° 51' 41" (2.8614°) | 1° 25' 50" (1.4307°) |
| Morse Taper MT3 | 0.6024 in/ft | 0.05020 in/in | 1 : 19.922 | 2° 52' 32" (2.8756°) | 1° 26' 16" (1.4378°) |
| Morse Taper MT4 | 0.6233 in/ft | 0.05194 in/in | 1 : 19.254 | 2° 58' 31" (2.9753°) | 1° 29' 15" (1.4876°) |
| Morse Taper MT5 | 0.6315 in/ft | 0.05263 in/in | 1 : 19.002 | 3° 00' 53" (3.0147°) | 1° 30' 26" (1.5074°) |
| CAT / BT / NMTB 7:24 Steep Taper | 3.5000 in/ft | 0.29167 in/in | 7 : 24 (1 : 3.429) | 16° 35' 39" (16.5943°) | 8° 17' 50" (8.2972°) |
| HSK Hollow Shank Taper (ISO 12164) | 1.2000 in/ft | 0.10000 in/in | 1 : 10.000 | 5° 43' 29" (5.7248°) | 2° 51' 45" (2.8624°) |
| ASME B1.20.1 NPT Pipe Taper | 0.7500 in/ft | 0.06250 in/in | 1 : 16.000 | 3° 34' 47" (3.5798°) | 1° 47' 24" (1.7899°) |
How Do You Calculate Lathe Tailstock Offset When Taper Length Differs from Total Shaft Length?
Turning long, shallow tapers (such as Morse tapers or arbors under 3°) by offsetting the lathe tailstock body allows longitudinal carriage power feeding. However, a frequent shop-floor mistake is calculating tailstock set-over using only the tapered section length (L_t) on a partially tapered shaft. Because the workpiece pivots between centers across its entire overall shaft length (L_w), the offset (O) scales by L_w / L_t:
Applying the exact tailstock set-over equation Offset O = [(D - d) / 2] × (L_w / L_t) = (TPI × L_w) / 2 = (TPF × L_w_in) / 24 guarantees that even when a short conical taper is machined on the end of a long drive shaft, the dead center lateral displacement scales accurately with the full distance between centers on the engine lathe bed without trial-and-error test cuts or wasted setup scrap.
| Taper Per Foot (TPF) | Taper Per Inch (TPI = TPF/12) | Included Angle 2α (Decimal°) | Compound Half-Angle α (Deg° Min' Sec") | Tailstock Offset for L = 254 mm (10.00") |
|---|---|---|---|---|
| 0.125 in/ft (1/8") | 0.01042 in/in | 0.5968° | 0° 17' 54" (0.2984°) | 1.323 mm (0.0521") |
| 0.250 in/ft (1/4") | 0.02083 in/in | 1.1937° | 0° 35' 49" (0.5968°) | 2.646 mm (0.1042") |
| 0.375 in/ft (3/8") | 0.03125 in/in | 1.7905° | 0° 53' 43" (0.8952°) | 3.969 mm (0.1563") |
| 0.500 in/ft (1/2") | 0.04167 in/in | 2.3873° | 1° 11' 37" (1.1937°) | 5.292 mm (0.2083") |
| 0.600 in/ft (Jarno/MT) | 0.05000 in/in | 2.8642° | 1° 25' 56" (1.4321°) | 6.350 mm (0.2500") |
| 0.750 in/ft (3/4" NPT) | 0.06250 in/in | 3.5798° | 1° 47' 24" (1.7899°) | 7.938 mm (0.3125") |
| 1.000 in/ft (1") | 0.08333 in/in | 4.7719° | 2° 23' 09" (2.3859°) | 10.583 mm (0.4167") |
| 1.500 in/ft (1-1/2") | 0.12500 in/in | 7.1527° | 3° 34' 35" (3.5763°) | 15.875 mm (0.6250") |
Standard Morse, Jarno, Brown & Sharpe, and 7:24 CAT/BT Spindle Taper Proportions
Per Machinery's Handbook (Industrial Press) and NIST (National Institute of Standards and Technology) metrology guides, Jarno tapers maintain a constant 0.6000 in/ft (exact 1:20 ratio) across all sizes, whereas Morse tapers (MT0–MT7) vary by size from 0.5986 in/ft (MT1) to 0.6315 in/ft (MT5). For high-accuracy taper inspection on a granite surface plate, combine this calculator with our Precision Sine Bar Gage Block Calculator, V-Block Cylinder Height Calculator, NPT Pipe Taper Tap Drill Chart, and Countersink & Chamfer Z-Depth Calculator.