110 Taper Angle Calculator

What the 110 in 110 taper angle means

An 110 taper angle is an included angle: the full opening between one side of the cone and the other, measured across the axis. Halve it and you get the half angle, 55 degrees, which is the angle between the cone surface and the centerline. That single distinction decides every number on this page, because the diameters D and d are measured across the full width, while the 55 degree angle is measured from the axis to one surface.

The other way to read 55 degrees is from the face of the part. A surface at 55 degrees to the axis sits at 35 degrees to the plane perpendicular to the axis, which is why a 110 degree cone looks like a shallow dish rather than a spike. A 60 degree lathe center, the classic example in shop training, is a 30 degree half angle and looks far sharper. A 118 degree drill point, the general purpose twist drill geometry, is a 59 degree half angle and is the closest common shape to the 110 degree cone.

The United States Army lathe manual TC 9-524, reproduced in the machining chapter hosted by Carnegie Mellon, states the rule plainly: the angle of the taper with the centerline is one half the included angle, and that is the angle the compound rest is set for. A 110 degree print therefore becomes a 55 degree setup, and everything downstream follows from that.

The three numbers that define any taper

Three measurements describe a straight taper: the large diameter D, the small diameter d, and the axial length L over which the change happens. Machinery's Handbook and every shop text build the same two ratios from them. Taper per inch is (D - d) / L, a pure number when both diameters and the length share a unit. Taper per foot is that value times 12, which is why American drawings often quote a taper as 0.75 in per foot instead of a ratio.

The angle comes from the same numbers. The radius shrinks by (D - d) / 2 across the length L, so the half angle is the arctangent of (D - d) / (2L), and the included angle is twice that. Written as one line: included angle = 2 x atan((D - d) / (2L)). The invert works too, and it is the form the calculator uses: D - d = 2 x L x tan(half angle).

For 110 degrees every term is fixed. The half angle is 55 degrees, its tangent is 1.4281, and the diameter change is 2 x 1.4281 = 2.8563 for every unit of length. Taper per inch is therefore 2.8563, taper per foot is 34.2756, and the taper ratio is 1 to 0.350.

Working the 110 degree case from one diameter

Take a small end of 0.500 in and a taper length of 1.000 in. The diameter change is 2 x 1.000 x 1.4281 = 2.8563 in, so the large end is 0.500 + 2.8563 = 3.3563 in. Check it backwards: 3.3563 minus 0.500 leaves 2.8563, halving that gives 1.4282, and the arctangent of 1.4282 lands within 0.002 degrees of 55, so the included angle returns to 110.000 within rounding. That round trip is the test that a taper calculation is consistent.

The same arithmetic in metric: a 20 mm small end over 25 mm of length gives 2 x 25 x 1.4281 = 71.41 mm of diameter change, so the large end is 91.41 mm. Nothing about the angle changes with the unit, because 2.8563 is a ratio of two lengths. The millimeter answer is the inch answer scaled by 25.4, which is a useful cross-check when a drawing mixes the two systems.

Where a 110 degree cone sits among standard tapers

Standard machine tapers are gentle by comparison. The Morse series, covered by ASME B5.10 and ISO 296, runs from 0.5986 to 0.6315 in per foot depending on size, which puts every Morse half angle between 1.43 and 1.51 degrees. Pipe threads are fixed by ASME B1.20.1 at a 1 in 16 taper, which is 0.0625 in per inch or 0.75 in per foot, a half angle of 1.7899 degrees. Brown and Sharpe tapers sit near 0.502 in per foot and Jarno tapers at exactly 0.600 in per foot.

The 110 degree cone is on a different scale entirely. At 2.8563 in per inch it is roughly 57 times the taper per inch of a Morse No. 2, and 46 times the taper per inch of an NPT thread. That is not a defect, it is a different job: shallow tapers align and grip tooling, while steep cones seat, center, chamfer and deburr. A 110 degree form tool or countersink removes a wide ring of material in a short axial travel, which is why the angle appears on deburring and chamfering tools rather than on drill shanks.

Setting a 110 degree taper on the lathe

The compound rest is the practical tool for this angle. Set it to 55 degrees from the axis, which is 35 degrees off the face, and feed with the compound rather than the carriage. The Army manual is explicit that the compound rest takes the half angle, so setting 110 on the dial would cut a cone twice as wide as the print asks for. On a lathe whose compound dial reads from the face instead of the axis, the number to set is 35 degrees, and the two readings differ by exactly 90 degrees.

Tailstock offset is the wrong method here, and the arithmetic shows why. Offset equals taper per inch times taper length divided by 2, so a 1.000 in long 110 degree taper would need 2.8563 x 1.000 / 2 = 1.4281 in of set-over. A typical engine lathe tailstock travels about 0.5 in either way, so the part cannot be held at that angle between centers at all. Taper attachments have similar limits: most are graduated for a few inches per foot, not 34.

That leaves three workable routes: a compound rest set to 55 degrees, a taper attachment with an extended guide bar, or a form tool ground to the profile. For short chamfers, the form tool or a 110 degree countersink is fastest. For a long conical seat, the compound rest gives the better surface finish because the tool keeps a constant nose radius relative to the cut.

Checking the angle with a sine bar

A sine bar turns the angle into a stack of gauge blocks, and the NIST SP 811 conversion table gives the degree to radian factor used inside the trigonometry: one degree is 1.745329E-02 rad. For a 5.000 in sine bar and a 55 degree half angle, the stack height is 5 x sin(55) = 4.0958 in. Set that stack under one roller, place the bar on the surface plate, and the top face now sits at 55 degrees.

Put the part on the bar with the taper surface against it and sweep a dial indicator along the length. A reading that stays within the tolerance band across the full travel confirms the angle; a steady rise or fall means the angle is off, and the size of the drift divided by the sweep length gives the error in radians. For a 2.000 in sweep, a 0.005 in drift is 0.0025 rad, about 0.143 degrees, which doubles to 0.286 degrees of included angle error. That is the same 2x relationship that links half angle to included angle, applied to a measurement instead of a print.

The Texas A&M taper turning notes cover the same sine bar method for shop use. The sine bar route also explains why 110 degrees is easy to check and hard to cut. The gauge stack of 4.0958 in is well within a normal block set, but the contact geometry is awkward: at 55 degrees the part tends to slide along the bar, and a stop or a clamp is needed. Most shops check a steep cone with a dial indicator on the machine instead, using the compound setting as the reference.

Degrees, ratio and taper per foot for the same cone

Three descriptions carry the same information, and moving between them is the part that trips people up. The angle is 110 degrees included, the ratio is 2.8563 to 1 of diameter change per unit length, and the taper per foot is 34.2756 in. Any one of them can be derived from the others: double the half angle, multiply the ratio by 12, or take twice the tangent of half the angle.

The table below puts the neighboring angles in the same frame, which is how you sanity check a value that looks wrong. A 90 degree countersink has a ratio of exactly 2.0000, because tan 45 is 1 and the diameter change is twice that. A 60 degree center comes in at 1.1547. If a calculation for 110 degrees lands anywhere below 2.5, the half angle was probably halved twice or the tangent was taken of 27.5 degrees.

Mistakes that break a 110 degree taper calculation

The most common error is feeding an included angle into a formula that expects a half angle. The arctangent of (D - d) / (2L) returns 55 degrees for a 110 degree cone, not 110, and a calculator that reports 110 from that expression has already doubled something it should not have. Check the direction: as the angle grows toward 180 degrees the cone flattens and the diameter change grows without limit, so a steep result belongs with a large angle.

The second error is forgetting the factor of two on the radius. Diameters shrink by twice what the radius does, so dropping the 2 halves the answer: 1.4281 instead of 2.8563. The third is mixing taper per inch with taper per foot. A Morse No. 2 quoted at 0.5994 is inches per foot, and treating it as inches per inch makes it 12 times too steep, which turns a 1.43 degree half angle into one near 16.7 degrees.

The fourth is assuming 110 degrees is a standard. It is not a machine taper in ASME B5.10 or ISO 296, so there is no gauge and no tolerance table behind it; the angle comes from the part print and the shop has to hit it directly. The last is unit drift inside one calculation: D and d in inches with L in millimeters gives a ratio off by 25.4, and the resulting angle will look plausible while being wrong by a wide margin. Convert everything to one unit before the arctangent, then convert the answer back if the print asks for it.