Work out how far to set the tailstock over to turn a taper between centers. Enter the two diameters, the tapered length, and the total length between centers — or a taper per foot — and get the set-over, the angle, and the taper rate.
Use this on a manual lathe without a taper attachment when you are turning a long, gentle taper on work held between centers — tapered shanks, tapered pins, or a tapered handle.
Measure the total length between centers of the actual workpiece, because the set-over tilts the whole part. If you are copying a standard taper, enter its taper per foot instead of diameters.
The tailstock offset method only works between centers; with one end in a chuck the result is not a true cone.
Offsetting the tailstock tilts the workpiece axis relative to the carriage travel. The cutting tool still moves parallel to the bed, so the part is cut to a cone whose half-angle matches the tilt. The tilt is the set-over divided by the full length between centers — which is why total length appears in the formula, not just the tapered length.
Diameter change per inch of length (TPI) is (D − d) divided by the tapered length. The set-over is half that diameter change spread over the whole part: total length times TPI over two. With a known taper per foot, that becomes total length times TPF over 24.
TPI = (D − d) / L_taper · TPF = TPI × 12
set-over = L_total × (D − d) / (2 × L_taper) = L_total × TPF / 24
half-angle = atan(TPI / 2) · included = 2 × half-angle
Each row below starts from the default inputs and changes only the columns shown. Recompute any row by hand with the formula above.
| D | d | L taper | L total | TPF in | Set-over | TPF | Included |
|---|---|---|---|---|---|---|---|
| 1.25 | 1 | 4 | 8 | 0 | 0.2500 | 0.7500 | 3.580° |
| 1.25 | 1 | 4 | 4 | 0 | 0.1250 | 0.7500 | 3.580° |
| 1.25 | 1 | 4 | 12 | 0 | 0.3750 | 0.7500 | 3.580° |
| 1.5 | 1 | 5 | 8 | 0 | 0.4000 | 1.2000 | 5.725° |
| 1.25 | 1 | 4 | 6 | 0.6 | 0.1500 | 0.6000 | 2.864° |
| 0.75 | 0.625 | 3 | 10 | 0 | 0.2083 | 0.5000 | 2.387° |
With the calculator defaults (Large diameter D (in) = 1.25; Small diameter d (in) = 1; Tapered length (in) = 4; Total length between centers (in) = 8; Taper per foot (optional) = 0), the arithmetic runs like this:
taper per inch = (D − d) / L_taper = (1.25 − 1) / 4 = 0.06250 in/in
set-over = L_total × (D − d) / (2 × L_taper) = 8 × 0.25 / (2 × 4) = 0.2500 in
half-angle = atan(TPI / 2) = 1.7899°; included = 3.5798°
Result: 0.2500 in set-over. Taper 0.7500 in/ft (0.06250 in per inch). Included angle 3.580°, half-angle 1.790°.
Now change Total length between centers (in) to 12 and leave everything else alone:
taper per inch = (D − d) / L_taper = (1.25 − 1) / 4 = 0.06250 in/in
set-over = L_total × (D − d) / (2 × L_taper) = 12 × 0.25 / (2 × 4) = 0.3750 in
half-angle = atan(TPI / 2) = 1.7899°; included = 3.5798°
Result: 0.3750 in set-over. Taper 0.7500 in/ft (0.06250 in per inch). Included angle 3.580°, half-angle 1.790°.
The same taper on a longer part needs proportionally more set-over: 0.25 in at 8 in between centers becomes 0.375 in at 12 in. Measure the real part, not the drawing’s tapered length.
Set the offset with a dial indicator on the tailstock quill, not by the witness marks on the tailstock base. Zero the indicator with the tailstock aligned, move it over by the set-over, and lock it before re-checking.
Center holes see the tilted axis and wear unevenly at large offsets. A ball-ended center helps, and so does keeping the set-over small — which is why steep tapers are usually cut with the compound rest instead.
Turn a test taper on scrap of the same length and measure both diameters before cutting the real part. Adjust the set-over by the ratio of the error and re-cut.
Units: Inches for diameters and lengths; taper per foot in inches per foot; angles in degrees.
Where the model stops: It assumes the work is held between two centers and the tool moves parallel to the bed. Center-hole depth and the small change in effective length at large offsets are not modeled.
Not a substitute: Use the lathe maker’s limits for tailstock travel and the drawing tolerance for the finished taper.
The tailstock tilts the entire workpiece. A short part and a long part with the same taper need different set-overs because the tilt angle is set-over divided by the full length between centers.
Then the tapered length equals the total length and the set-over is simply half the diameter change. A 1.25 to 1.00 in taper over the full part needs 0.125 in.
Yes, between centers, if your tailstock has enough travel. Enter the standard’s taper per foot from a reference table and the length between centers.
It depends on the lathe. Many small lathes allow only a fraction of an inch, and large offsets also wear the center holes. For steep tapers use the compound rest or a taper attachment.