The mistake that ruins most taper setups
A taper is specified in diameters and cut along a radius. Everything that goes wrong in taper turning comes from mixing the two up.
If a part goes from 1.000 to 0.750 inch over 3 inches, the diameter changed by 0.250. But the surface the tool follows only moved 0.125 — half of it, because the other half came off the far side. So the angle of that surface is arctan(0.125 ÷ 3) = 2.386 degrees, not arctan(0.250 ÷ 3) = 4.76.
That gives two numbers, both correct and both used. The included angle is the full angle across the taper, 4.772 degrees here, and it is what a drawing usually calls out and what you would check with a sine bar. The half angle is the angle of one flank, 2.386 degrees, and it is what you set on the compound rest — because the compound travels along the surface being cut, not across the axis.
Setting the compound to the included angle produces a taper exactly twice as steep as intended. It is the classic error, and it is worth checking which number you are holding before touching the machine.
Three ways to cut a taper, and when each one works
The compound rest is the most direct: swivel it to the half angle and feed by hand. It cuts any angle up to the limit of the rest's swivel, works on a part held any way, and needs no accessories. Its limitation is travel — the compound is short, so long tapers cannot be cut in one pass, and hand feeding shows in the finish.
Tailstock offset tilts the whole workpiece by moving the tailstock sideways, then cuts with the ordinary power longitudinal feed. That gives long tapers and a good finish. But it only works between centres, it only suits shallow tapers, and the offset depends on the overall length of the work rather than the length of the taper. The same taper on a longer bar needs a bigger offset.
A taper attachment guides the cross slide from a separate angled bar. It gives power feed, long tapers, no disturbance to the centres, and works on chucked or centred work. It is simply another accessory to own and set.
There is a real defect specific to the offset method: because the centres are no longer aligned with the centre holes, they bear on one edge rather than seating properly. On any significant offset that produces a slightly curved taper and chews up the centre holes. Keep offsets small, and use one of the other two methods when the angle gets steep.
Worked example: 1.000 to 0.750 over 3 inches
An 8-inch bar held between centres, with a 3-inch taper running from 1.000 down to 0.750 inch diameter.
- Diameter difference. 1.000 − 0.750 = 0.250 inch.
- Taper per inch. 0.250 ÷ 3 = 0.08333 in/in, so taper per foot is 0.08333 × 12 = 1.000 in/ft — a one-inch-per-foot taper, which is a common callout.
- Half angle. Halve the difference first: 0.125. Then arctan(0.125 ÷ 3) = 2.3859°. This is the compound setting.
- Included angle. Double it: 4.7719°. This is what a drawing calls out and what you would verify.
- Tailstock offset. Half the taper per inch is 0.041667; multiply by the overall 8 inches to get 0.3333 inch.
Note what the offset depends on. Cut the identical 3-inch taper on a 12-inch bar and the offset becomes 0.500 inch, not 0.333 — the tilt is applied to the whole part, so the longer the part, the further the tailstock has to move to achieve the same slope.
Standard machine tapers, and why they are all shallow
Machine tapers exist to hold tools by friction, and that requires a shallow, self-holding angle. A Morse taper runs about 0.0520 inch per inch, which is roughly 0.625 inch per foot and an included angle near 2.98 degrees. Brown & Sharpe is shallower still at about 0.500 inch per foot.
Below roughly 3 degrees included, a taper is self-holding: friction on the flanks resists the axial force trying to push it out, so the tool stays put with no drawbar. That is why drill chucks and centres use Morse tapers and why they need a drift to remove.
Steeper tapers are self-releasing. The R8 taper at 0.125 inch per inch, and the 7/24 tapers used on milling machine spindles, will not hold themselves and are always retained by a drawbar. The advantage is that they come out cleanly and repeatably, which matters far more on a machine that changes tools often.
Morse tapers are also not quite constant across sizes. The nominal figure is a useful approximation but each number has its own slightly different rate, so cut a replacement from the published dimension for that size rather than from the nominal.
Common taper callouts
| Taper | Per inch | Per foot | Included angle | Holding |
|---|---|---|---|---|
| Brown & Sharpe (nominal) | 0.04165 | 0.500 | 2.386° | Self-holding |
| Morse (nominal) | 0.05205 | 0.625 | 2.981° | Self-holding |
| Jarno | 0.05000 | 0.600 | 2.864° | Self-holding |
| 3/4 inch per foot | 0.06250 | 0.750 | 3.576° | Marginal |
| 1 inch per foot | 0.08333 | 1.000 | 4.772° | Self-releasing |
| R8 (Bridgeport) | 0.12500 | 1.500 | 7.153° | Drawbar |
Morse and Brown & Sharpe rates vary slightly by size number. Use the published dimensions for the specific taper when cutting a replacement part.
The compound takes the half angle
Set the compound rest to the half angle, not the included angle. The compound feeds along one flank of the cone, so it needs the angle of that flank. Using the included angle doubles the taper. If a taper comes out roughly twice as steep as intended, this is almost certainly why.
Getting a taper right
- Know which angle you are holding. Included for the drawing and for inspection; half for the compound rest.
- Use the axial length, not the slant length. The formula wants the distance along the axis between the two diameters.
- Remember the offset depends on overall length. A longer workpiece needs a larger tailstock offset for the identical taper.
- Check the offset against the centre holes. A large offset makes the centres bear on one edge, which curves the taper and damages the holes.
- Cut, measure, adjust. Take a light pass, measure two diameters a known distance apart, and recompute. Small setting errors are far easier to correct than to predict.
Checking the taper you cut
Measuring a taper is a separate job from cutting one. The reliable method is to measure two diameters a known axial distance apart and put them back through this calculation — if the taper per inch matches, the angle is right, and it needs no angular measurement at all.
Where you do need to verify the angle directly, the sine bar calculator gives the gauge block stack to set the part at its nominal included angle so an indicator can sweep the flank flat. For fits on the finished diameters, the hole and shaft fit calculator covers the tolerance question, and the cutting speed calculator handles the fact that surface speed changes continuously along a taper — the small end is turning much slower in surface terms than the large end.
Terms used here
- Included angle
- The full angle across the taper, between the two opposite flanks. What drawings normally specify.
- Half angle
- The angle of one flank from the centreline. Half the included angle, and the number the compound rest is set to.
- Taper per foot
- Change in diameter, in inches, per foot of axial length. Twelve times the taper per inch.
- Self-holding taper
- A taper shallow enough that friction retains the tool without a drawbar, roughly below 3 degrees included. Morse and Brown & Sharpe are self-holding; R8 is not.