How this instrument works
A taper is a workpiece that changes diameter steadily along its length — a shaft, a pin, a reamer or a machine spindle that's wider at one end than the other by design, not by mistake. Machinists describe how steep that taper is in two different, equally standard ways: taper per foot, a shop-floor figure that says how many inches the diameter would change over a full 12-inch length if the taper continued at the same rate, and the included half-angle, the actual geometric angle between the workpiece's centerline and its tapered surface.
Taper per foot comes from a simple scaling: take the actual diameter difference over the actual length, then scale that ratio up to what it would be over a full foot by multiplying by 12. It's a rate, not a real physical dimension on the part — a 6-inch-long taper and a 12-inch-long taper can share the identical taper-per-foot figure even though one is twice as long, as long as their diameter change is scaled proportionally.
The included half-angle is different: it's the actual angle a lathe's compound rest needs to be set to in order to cut the taper, found from the diameter difference and twice the length (since the radius, not the diameter, is what actually forms the angle from the centerline). Machinists use taper per foot for quick shop-talk and drawing callouts, but they dial the compound rest to the half-angle in degrees, since that's the number a protractor scale on the machine actually reads.
- Enter the wider end's diameter into Large diameter (in).
- Enter the narrower end's diameter into Small diameter (in).
- Enter the distance between them into Taper length (in) — the length over which the diameter actually changes.
- Read Taper per foot (in/ft) for the shop-standard rate figure, often used in drawing callouts.
- Read Taper angle (included half-angle, deg) for the angle to dial into a lathe's compound rest when cutting the taper.
Worked example — 2-inch to 1-inch diameter over 6 inches
Enter 2 into Large diameter (in), 1 into Small diameter (in), and 6 into Taper length (in). Taper per foot (in/ft) reads 2.0000 in/ft, and Taper angle (included half-angle, deg) reads 4.7636°.
By hand: the diameter changes by 2 − 1 = 1 inch over 6 inches of length, a rate of 1/6 in per inch; scaled up to a foot, that's (1/6) × 12 = 2.0 in/ft exactly. The half-angle comes from the radius change instead: 1 divided by (2 × 6) = 1/12, and the arctangent of 1/12 is approximately 4.7636°, independently checkable on any scientific calculator's inverse-tangent function.
Questions
What's the difference between taper per foot and the included half-angle?
Taper per foot is a shop-floor rate — how much the diameter would change over a scaled-up 12-inch length — commonly used in drawing callouts and quick verbal shop-talk. The included half-angle is the actual geometric angle between the workpiece's centerline and its tapered surface, and it's the number a lathe's compound rest protractor scale actually reads when setting up to cut the taper. Both describe the same physical taper, just in different units for different purposes.
Why does the half-angle formula divide by 2 × length but taper per foot doesn't?
Because the half-angle is measured from the workpiece's centerline out to its surface, which uses the radius change, not the diameter change — and the radius change is exactly half the diameter change. Taper per foot describes the diameter's rate of change directly, with no centerline reference needed, so it uses the full diameter difference without halving it.
Is taper per foot the same regardless of how long the tapered section is?
It can be, as long as the ratio of diameter change to length stays constant — a 3-inch taper section and a 6-inch taper section can share an identical taper-per-foot figure if the second one's diameter change is exactly double the first's. Taper per foot is a rate (inches of change per foot of length), not a fixed physical dimension of any particular part.
What's a common example of a real-world taper in machining?
Morse tapers, used on drill shanks, lathe centers and tool holders, are a widely used standardized family of self-holding tapers with published taper-per-foot and angle figures for each numbered size. A drawing calling out a taper by its taper-per-foot figure (like 5/8 inch per foot, a common Morse taper rate) is describing exactly the kind of ratio this instrument computes from two diameters and a length.
Does the large diameter have to be at a specific end of the workpiece?
No — the formula only needs to know the two diameters and the length between them; it doesn't care which physical end each diameter sits at. As long as Large diameter (in) is entered as the bigger of the two numbers and Small diameter (in) as the smaller, the resulting taper per foot and half-angle describe the taper correctly regardless of orientation.