Lathe Taper & Tailstock Offset Calculator
Enter the finished end diameters and taper length to set a compound, plan a tailstock offset, and check a diameter partway along the work.
Compound-rest angle from spindle axis2.3859°
- Included taper angle
- 4.7719°
- Taper per in
- 0.0833 in/in
- Taper per foot
- 1.0000 in/ft
- Taper ratio
- 1:12.0000 (diameter decreases from the stated large end)
- Conventional tailstock set-over
- 0.3333 in
- Geometry-corrected set-over
- 0.3330 in
- Approximation difference
- 0.000289 in (0.0868%)
- Diameter at inspection position
- 1.1250 in
Tailstock set-over is generally for long, shallow tapers turned between centers; make a trial cut and inspect the result.
How to use this calculator
- Choose inches or millimeters, then use that unit for every dimension in the form.
- Enter the finished large and small diameters and the axial length over which they change.
- Enter the full distance between centers when you need a tailstock set-over value.
- Enter the measurement location from the small end to check the expected diameter there.
How the taper and set-over are calculated
Let D be the large finished diameter, d the small finished diameter, l the tapered axial length, and L the distance between centers. The diameter taper per unit length is:
T = (D − d) ÷ l
The compound-rest setting is the taper half-angle, measured from the spindle axis rather than from the cross-slide:
α = atan((D − d) ÷ (2l))
The included angle is 2α. A taper ratio is shown as 1:n, where n = l ÷ (D − d). For inch taper per foot, the calculator multiplies the dimensionless taper by 12: TPF = 12(D − d) ÷ l. That conversion is valid with metric input too because millimeters per millimeter and inches per inch have the same numerical ratio.
Conventional and geometry-corrected tailstock set-over
The familiar shop set-over formula is Oa = L(D − d) ÷ (2l), or LT ÷ 2. It is a small-angle approximation. The calculator also gives the geometric center displacement Oe = L sin(α), using the same half-angle that sets the compound. The difference between them is usually very small on a shallow taper, but it becomes more visible as the angle grows.
Use L for the complete length between the live and dead center points, not just the tapered portion. The tapered length controls the taper itself; the distance between centers controls how far the tailstock must move to point the supported work at that angle. A negative result means the entered diameters describe a taper in the opposite direction, and the note identifies that direction.
Checking a diameter along the taper
At a distance x from the small end, the expected diameter is Dx = d + x(D − d) ÷ l. This is useful for a micrometer check after a trial cut. A position beyond the stated taper length is calculated as a straight-line extrapolation and is flagged rather than treated as a real inspection point.
What this calculator leaves out
Tailstock set-over is normally for long, shallow tapers turned between centers. This calculation does not know the lathe's set-over travel, center-hole condition, bed alignment, tool deflection, cutting force, thermal movement, or measuring error. A steep taper can be mathematically valid but impractical for a tailstock setup; use the machine manual, make a trial cut, and choose a compound or taper attachment when the setup calls for it.
Worked example
For a taper from 1.2500 in down to 1.0000 in over 3.0000 in, with 8.0000 in between centers, the diameter change is 0.2500 in. The taper is 0.2500 ÷ 3.0000 = 0.083333 in/in, or 1.0000 in/ft, and the ratio is 1:12.
The half-angle is atan(0.2500 ÷ 6.0000) = 2.3859°, so set the compound to 2.3859° from the spindle axis, not to the 4.7719° included angle. The conventional set-over is 8 × 0.2500 ÷ 6 = 0.333333 in; the geometry-corrected result is 0.333044 in. At 1.5000 in from the small end, the expected diameter is 1.1250 in.
Common questions
Why is the distance between centers different from the taper length?
Taper length is only the axial distance over which the diameter changes. Distance between centers is the full supported span of the work, and that span determines how much the tailstock must move to aim the work at the taper angle. They may be equal, but often there is straight material outside the taper.
Do I set the compound to the included angle or the half-angle?
Set the compound to the half-angle from the spindle axis. The included angle is twice that value because it includes both sides of the taper. Setting the compound to the included angle would produce a taper that is much too steep.
How is tailstock set-over calculated?
The common approximation is distance between centers times the diameter change, divided by twice the taper length: L(D − d) ÷ 2l. This calculator also shows L sin(α), which uses the calculated half-angle and is the geometric center displacement.
Why does the exact offset differ from the common shop formula?
The common formula replaces the sine of the taper half-angle with the angle's small-angle approximation. At shallow angles the two answers are nearly the same, which is why the conventional formula works well in ordinary tailstock work. The difference grows as the taper gets steeper.
When should I use a compound rest instead of tailstock offset?
Use tailstock offset mainly for long, shallow external tapers between centers. A compound is usually better for short or steeper tapers, and a taper attachment is useful when the work cannot be offset or when parallel features must remain aligned. Machine travel, rigidity, and the drawing decide the final method.
What does a 1:10 taper ratio mean?
A 1:10 taper changes diameter by one unit over ten units of axial length. For example, it changes 0.100 in in 1.000 in of length, or 1 mm in 10 mm. The ratio describes diameter change, not radial change.