Taper Calculator

What a taper is and the three numbers that describe it

A taper is a shape whose diameter changes at a steady rate from one end to the other. A funnel, a wooden chair leg, the shank of a drill bit, and the threaded end of a pipe are all tapers. The math that describes them is the same in every case, and it starts with three measurements.

The large diameter (D) is the wide end. The small diameter (d) is the narrow end. The taper length (L) is the distance between those two ends measured along the centerline, not along the sloped surface. Those three numbers fix the shape, so everything else on this page is a short step away from them.

Two words get confused a lot. A taper is a change in diameter along the length of a round part. A bevel is a cut across an edge. A chamfer is a small bevel at the mouth of a hole. You can cut a taper on a lathe with a compound rest or a taper attachment, and you can cut a bevel with a saw or a router. The formulas below are for tapers only.

The taper formulas, step by step

All of the numbers come from one right triangle. Slice the taper down the middle and you get a triangle whose short side is half the diameter drop, (D - d) / 2, and whose long side is the taper length L. The angle at the tip is the half angle.

That gives the working set of formulas:

taper per inch = (D - d) / L

taper per foot = 12 x taper per inch

half angle = atan((D - d) / (2L))

included angle = 2 x half angle

The factor of 2 in the angle formula is the part people drop. Diameters are measured across the whole circle, but the triangle only uses half of that drop, so the drop gets divided by 2 before the arctangent. Skip that step and the angle comes out roughly twice as large as it should be.

The formula appendix of US Army training circular TC 9-524 lists the same rules in shop language: subtract the small diameter from the large one, divide by the length of the taper, and multiply by 12 for taper per foot.

Worked example: a 1.250 in end down to 0.625 in over 3.000 in

Take a shaft that measures 1.250 in at the large end and 0.625 in at the small end, with 3.000 in of taper between them. The diameter drop is 1.250 - 0.625 = 0.625 in.

Taper per inch is 0.625 / 3.000 = 0.2083 in per inch. That means the diameter shrinks by about 0.208 in for every inch you travel along the axis. Taper per foot is 12 times that, or 2.50 in per foot, which is the number most drawings quote.

The half angle is atan(0.625 / 6.000) = atan(0.10417) = 5.95 degrees. The included angle is twice that, 11.89 degrees, or 11°53'37.4". The ratio form is 1 to 4.8.

The Army lathe example: 22 degrees 37 minutes

The taper turning chapter of TC 9-524, chapter 7 works a classic example that is worth repeating because it shows how the half angle and the included angle differ in practice. The part is 1.000 in at the large end, 0.375 in at the small end, and 0.750 in long.

The diameter drop is 0.625 in. Taper per inch is 0.625 / 0.750 = 0.8333, and taper per foot is 10.0. The half angle is atan(0.8333 / 2) = 22.62 degrees, which is 22 degrees 37 minutes. The included angle is twice that, 45.24 degrees, or 45 degrees 14 minutes.

The manual then makes the point that matters at the machine: the compound rest is set to the half angle, 22 degrees 37 minutes, not to the 45 degree included angle. The compound rest swings around a vertical axis, so its scale reads the angle between the tool path and the lathe centerline. Setting it to the included angle would cut a cone that is twice as steep as the drawing.

That example also explains why machinists talk in degrees and minutes. A lathe scale marked in whole degrees cannot show 37 minutes, so the manual tells the operator to set the nearest degree, take a trial cut, measure, and adjust. The calculator on this page gives both the decimal angle and the degrees, minutes, seconds form so you can compare with either kind of scale.

Taper per foot, taper per inch, and degrees are three views of one cone

Drawings in the United States usually give a taper as inches per foot, because that is easy to check with a ruler over a 12 in length. A machinist may prefer inches per inch, because the lathe carriage moves in inches. A inspector may prefer degrees, because a sine bar or a protractor reads in degrees. All three describe the same cone, and the table below converts between them.

The conversion from taper per foot to angle is half angle = atan(TPF / 24). The 24 comes from two divisions: divide by 12 to get the drop per inch, then divide by 2 because the triangle uses half the diameter drop. The included angle is twice the half angle.

Notice that the angle does not grow in a straight line with taper per foot. Going from 0.25 to 0.50 doubles the taper per foot and doubles the angle, because the angles are small. Going from 2.0 to 3.0 raises the taper by half again but the included angle only moves from 9.53 to 14.25 degrees. The tangent curve flattens out, so at steep tapers a small change in angle means a large change in taper per foot.

Pipe threads are a familiar case. NBS Handbook H28, the federal screw thread standard, specifies American Standard taper pipe threads at a taper of 1 in 16 measured on the diameter. That is 0.0625 in per inch, 0.75 in per foot, and an included angle of 3.58 degrees. The shallow angle is what lets a pipe fitting seal as it is tightened.

Tailstock offset and the compound rest: two ways to cut the taper

A lathe can cut a taper in more than one way, and each method uses a different number from the list above.

The compound rest method swings the tool at the half angle and feeds it by hand along the slope. It works for short tapers and for internal tapers. The catch is that the travel of a compound rest is short, often 3 to 5 in, so long tapers do not fit.

The tailstock offset method shifts the tailstock sideways so the workpiece sits at an angle between centers. The tool then travels parallel to the bed and cuts a taper over the full length. The offset is half the diameter drop, scaled by how much longer the workpiece is than the tapered section:

tailstock offset = ((D - d) / 2) x (total length / taper length)

For a shaft that goes from 2.000 in to 1.500 in over 4.000 in, with 12.000 in between centers, the offset is (0.500 / 2) x (12.000 / 4.000) = 0.250 x 3 = 0.750 in. The same answer comes from the taper per foot form in the manual, TPF x total length / 24, which gives 1.5 x 12 / 24 = 0.750 in. Both routes are in the calculator.

Where tapers show up in a shop and around the house

Tapers are everywhere once you start looking. Drill chucks and lathe centers use Morse tapers, a family of self-holding shanks that grip by friction and release when a wedge is tapped in. The Jarno taper is unusual because every size in the family uses the same taper, 0.600 in per foot, so the angle never changes even when the diameter does.

Pipe threads are tapered so the joint tightens and seals as the fitting is turned. The standard calls for 1 in 16 on the diameter, as noted above. A tapered thread is also why you should not force a fitting past snug: the threads are doing the sealing, and over tightening can split the fitting.

Woodworkers cut tapers on legs, dowels, and tool handles. A chair leg that goes from 1.5 in at the top to 1.0 in at the floor over 16 in has a taper per inch of 0.0313 and a half angle of 0.90 degrees, a subtle line that reads as a shaped leg rather than a cone.

Common mistakes and how to check your taper

Most wrong taper answers come from one of five slips, and each one leaves a fingerprint you can spot. The table below lists them with the fix.

The first mistake is the most common. Taper is defined on the diameter, so a part that drops 0.500 in on the radius has dropped 1.000 in on the diameter. Use the full diameter at each end, not the distance from the centerline to the surface.

The second is measuring the length along the slope with a tape or a rule. For a shallow taper the difference is tiny, but for a steep one it matters. Hold the rule parallel to the centerline, or measure between shoulders that are square to the axis.

To check a finished taper, set a dial indicator on the tool post, touch the surface, and move the carriage a known distance along the axis. The indicator reading divided by the travel is the taper per inch. A second check is to measure the diameter at two points a known distance apart with a micrometer. For a short part, a sine bar and gage blocks give the angle directly: the height of the gage block stack is the sine of the angle times the bar length, a rule also listed in the TC 9-524 appendix.