Turning Tapers

A taper changes diameter at a constant rate

A straight taper is a cone, or part of one. Name the large diameter D, small diameter d and axial length of the tapered portion l. Use the same length unit for all three.

Taper rate = (D − d) / l. With inch dimensions, this is taper per inch, often abbreviated TPI. Here TPI means diameter change per inch of length; in Single-Point Threading, the same abbreviation means threads per inch. Write the units to distinguish them.

TPF = 12 × TPI, and TPI = TPF / 12. Taper per foot describes the diameter change over twelve inches; the actual tapered portion need not be a foot long.

Check the rate with a fresh example

A taper goes from 1.100 to 0.950 in over 3.000 in. Its diameter changes 0.150 in, so the rate is 0.050 in/in, or 0.600 in/ft. At half the length, the diameter has changed half as much: 0.075 in. These are teaching dimensions, not a standard machine taper.

Set the compound to the half-angle measured from the spindle axis

The tool travels along one side of the cone. Its angle to the spindle axis is the half-angle α; the angle between the two sides is the included angle 2α.

α = atan((D − d) / (2l)) = atan(TPI / 2). Use degree mode for a setting in degrees. A rate of 0.050 in/in gives α = 1.432°, while the included angle is 2.864°.

Swivel the compound so its travel makes that half-angle with the spindle axis, then feed along the compound. The taper length is limited by compound travel. Check how the lathe labels its scale: a scale measured from cross-slide travel uses the complementary angle, 90° − α.

Why the factor of two belongs in the angle formula

Cut the cone through its axis. One side forms a right triangle whose rise is (D − d)/2 and whose run is l. Tangent is rise divided by run. Using the whole diameter change as the rise gives the wrong tool path.

Tailstock offset uses the full supported work length

For a shallow external taper turned between centers, offsetting the tailstock tilts the work while the carriage still travels parallel to the bed. The usual small-angle setup estimate is offset ≈ TPI × L / 2 = (D − d)L / (2l).

L is the total work length between the effective center support points, approximated by the total part length in elementary problems. It is not just the length l of the tapered portion. Center-hole geometry matters in an actual setup. Use inches for L when the taper rate is in in/in; if given TPF, first divide it by 12.

For a rate of 0.050 in/in and L = 8.000 in, the estimate is 0.200 in even if only 3.000 in is tapered. Machine offset capacity and the center arrangement limit this method. Check a trial cut and restore center alignment before straight turning.

Why this is an estimate rather than an exact setting

The taper gives tan(α) = TPI/2. If L is the actual distance along the tilted work between support points, the ideal geometric offset is L sin(α). At shallow angles sin(α) and tan(α) are nearly equal. The usual formula substitutes L tan(α); real center contact and setup error add further differences. It is not a method for steep cones.

Choose the taper method for length, access and support

Compound rest

Useful for short external or internal tapers. The compound guides the tool at the half-angle, but its limited travel restricts length.

Tailstock offset

Useful for shallow external tapers between centers. It changes the work axis and does not provide a method for boring an internal taper.

Taper attachment

A guide coordinates cross-slide motion with carriage travel. It can produce internal or external tapers within its travel and angle limits while the work stays aligned with the spindle. Connection and cross-feed arrangements depend on the attachment.

Form tool

A shaped cutting edge produces a short taper by feeding across the axis. A wide edge engages much of the cut at once, increasing cutting load and the tendency to chatter.

Start with Lathe Operations for the machine and workholding. Use Feeds and Speeds for cutting conditions and Surface Texture and Finish for the finish left by the cut.

A matching taper needs both the right slope and the right size

Measure diameters at two known axial positions to check the rate. A correct rate can still leave the entire taper too large or too small. A suitable taper plug or ring gauge checks the mating geometry; contact pattern and seating position answer different questions.

Morse tapers are commonly used for centers and tool shanks. Jacobs tapers commonly join a drill chuck to its arbor. They are separate families: identify the exact mating designation before selecting a tool or adapter. No dimensions or angles should be inferred from the family name alone.

No standard taper table is reproduced here. Use the specified component drawing and verified manufacturer or standard dimensions for a real fit; see Tolerances and Fits.

Practice taper rate Practice half-angle Practice offset