Taper Calculator — taper per foot, angle and set-over

A taper can be described four different ways — as two diameters over a length, as a taper per foot, as an angle from the centreline, or as a ratio — and drawings, tooling catalogues and machinists all pick different ones. This taper calculator moves between them in either direction, and adds the two numbers you actually need at the machine: the compound rest angle and the tailstock set-over.

Arb Digital publishes free workshop and construction calculators, and this one exists because the conversions hide two easy mistakes. Taper per foot is a change in diameter, so the angle you set on a compound rest is derived from half of it. And set-over depends on the total length of the part, not on the length of the tapered section, so two parts with identical tapers need different set-overs if their overall lengths differ.

What This Taper Calculator Does

Give it any three of the large diameter, the small diameter, the tapered length and the taper per foot, and it solves for the fourth. Alongside that it reports the included angle across the full taper, the angle measured from the centreline, the taper expressed as a ratio in the form 1 in n, and the tailstock set-over for turning the taper between centres on a lathe.

Four solve modes cover the situations that actually come up. You have measured a part and want its taper per foot. You have a taper specification and a large end and need the small end. You are working back from a small end. Or you know both ends and the taper and need to know how long the tapered section has to be.

The unit selector switches every length field between inches and millimetres. The angles and the ratio are unit-free and mean the same thing either way, which is why they travel better across drawings from different sources than a taper-per-foot figure does.

How to Use It

1. Pick the solve mode that matches what you know. The field being solved for is ignored, so leaving an old value in it does no harm.
2. Enter diameters, not radii. Measure square across the part at each end of the tapered section, not at the extreme ends of the workpiece if there is parallel material beyond the taper.
3. Measure length along the axis. The distance along the sloping surface is slightly longer, and using it makes the taper come out shallower than intended.
4. Add the total length between centres if you want the set-over figure. If the whole part is tapered, it is the same as the tapered length.
5. Read the angle from the centreline for the compound rest, and the included angle when comparing against a drawing that specifies the full angle.

The Formula and How It Is Calculated

Every output comes from one quantity: the change in diameter over the tapered length. Taper per inch is (D − d) ÷ L, and taper per foot is that multiplied by twelve. Going the other way, the diameter change is TPF × L ÷ 12, which is how the small and large end modes work.

The angles come from a right triangle whose opposite side is half the diameter change, because the surface slopes away from the centreline on both sides. The angle from the centreline is arctan((D − d) ÷ 2L), and the included angle is twice that. This factor of two is the single most common source of error in taper work.

The taper ratio is L ÷ (D − d), reported as 1 in n. A ratio of 1 in 10 means the diameter changes by one unit for every ten units of length, and it is the notation used for many standard tapers and for pipe threads.

Tailstock set-over is total length × (D − d) ÷ (2 × tapered length). Offsetting the tailstock tilts the whole workpiece relative to the carriage travel, so the taper it produces is set by the angle of the part as a whole — which is why the total length appears in the formula and the tapered length only appears as the reference the taper was specified over.

Worked example: a part with a 1.5 inch large end, a 1.0 inch small end and a 5 inch tapered section. The diameter change is 0.5 inches, so the taper per inch is 0.1 and the taper per foot is 1.2 inches. The angle from the centreline is arctan(0.25 ÷ 5) = arctan(0.05) = 2.862°, and the included angle is 5.725°. The ratio is 5 ÷ 0.5 = 1 in 10. If the whole part is 8 inches between centres, the set-over is 8 × 0.5 ÷ 10 = 0.4 inches.

Why Set-Over Depends on the Whole Part

This trips up people who have only ever used the compound rest. When you offset the tailstock, you are not adjusting a tool angle — you are tilting the workpiece. The carriage still travels parallel to the bed, so the taper produced is the angle the part now sits at, and that angle is the set-over divided by the distance between the centres.

The practical consequence is that a specified taper needs a different set-over on a long part than on a short one. Turn a 1 in 10 taper on a 4 inch part and the set-over is half what it is on an 8 inch part with the same taper. Get it the wrong way round and the part comes off the machine with a taper twice or half what the drawing asked for, and no amount of care with the cut will fix it.

Set-over also has a drawback the compound method does not: because the centres are no longer aligned with the part's axis, the centre holes bear on one side and wear unevenly, which limits the accuracy achievable and the amount of offset that is sensible. That is a machining judgement rather than a calculation, and it belongs with the machinist and the machine's own limits.

Standard Tapers and Why They Are Quoted Differently

Machine tool tapers fall into two families. Self-holding tapers, such as the Morse series, are shallow enough that friction retains the tool without a drawbar. Steep or self-releasing tapers are much sharper and rely on a retention mechanism, which makes them faster to change. Both families are dimensioned in published standards rather than being free choices, and the geometry is specified in ASME B5.10, Machine Tapers — Self Holding and Steep Taper Series.

Because those standards exist, this page publishes no table of standard taper sizes. If you are matching an existing spindle or shank, take the figures from the standard or from the machine's documentation rather than from a calculator's memory. What this tool is for is the general conversion — turning whatever numbers you were given into the form your machine or drawing needs. Measurement traceability and the units themselves are the province of national metrology bodies such as the NIST Office of Weights and Measures.

Tapers Outside the Machine Shop

The same arithmetic covers a lot of work that has nothing to do with a lathe. A tapered column, a tapered leg on a table, a tapered post, a wedge, a conical hopper and a chamfer on a large casting are all described by two diameters or widths and a length. Woodworkers usually think in terms of the angle, metalworkers in taper per foot, and civil drawings often use the ratio — but they are the same number in three costumes.

For flat rather than round work the same relationship applies with widths instead of diameters, with one caveat: on a flat taper cut on one face only, the full change happens on one side, so the angle is arctan of the whole change over the length rather than half of it. Keeping track of whether a taper is one-sided or symmetrical about an axis is worth more than any formula. The miter angle calculator covers the related problem of setting a saw for a corner rather than a taper.

Measuring an Existing Taper

To reverse-engineer a taper on a finished part, measure the diameter at two points a known distance apart, along the axis, and enter those as the large and small ends with that distance as the tapered length. Pick the two measuring points as far apart as the taper allows: measuring over a short span magnifies any error in the diameters into a large error in the angle.

A worked check: if two diameters 0.5 inches apart differ by 0.05 inches, the taper per foot is 1.2 inches — the same as the example above — but a thousandth of error in either diameter shifts that result by more than two per cent. Over five inches the same thousandth is almost invisible. If the part is round and you need related geometry, the circle calculator and the bolt circle calculator cover the layout side, and the material removal rate calculator covers what happens once you start cutting.

Common Mistakes to Avoid

• Setting the compound to the included angle. The compound rest is set to the angle from the centreline, which is half the included angle. Getting this wrong doubles the taper.
• Using the tapered length for set-over. Set-over depends on the total distance between centres. A short taper on a long part needs more offset than the tapered length alone implies.
• Measuring length along the surface. The axial length is what every formula here uses; the sloping surface is slightly longer and produces a shallower taper than intended.
• Confusing taper per foot with taper per inch. They differ by a factor of twelve, and both notations appear on drawings without always being labelled.
• Treating a one-sided taper as symmetrical. On a flat taper cut on a single face, the whole dimensional change happens on that face, so the angle is not halved.

Related Free Tools From Arb Digital

Derive thread geometry with the thread pitch calculator, size fastener holes with the clearance hole calculator, set a saw with the miter angle calculator, lay out round work with the circle calculator or the bolt circle calculator, and check cutting rates with the material removal rate calculator. Everything else is in the free tools hub.

Frequently Asked Questions

This tool performs geometric conversions from the dimensions you enter. Standard taper sizes, tolerances, fits and machining practice are set by published standards and by the machine and tooling manufacturers; confirm any figure against the relevant standard or documentation before cutting.