How it works
A taper changes diameter along its length. Taper per foot is the diameter change over one foot of length. Because the diameter changes equally on both sides, the half angle (the angle each side makes with the axis) is arctan of half the diameter change over the length, and the included angle is twice that.
Machinists set a lathe compound or tailstock offset from the half angle or from the taper per foot. Standard tapers such as Morse and Jacobs have published dimensions; use those for interchangeable parts.
Worked example
A part tapering from 1.000 in to 0.750 in over 3 in:
- Taper per foot = (1 − 0.75) ÷ 3 × 12 = 1 in
- Half angle = arctan((1 − 0.75) ÷ (2 × 3)) = 2.3859°
- Included angle = 2 × 2.3859° = 4.7719°; ratio = 3 ÷ 0.25 = 12 : 1
| Input | Value |
|---|---|
| Large diameter | 1 in |
| Small diameter | 0.75 in |
| Length of the taper | 3 in |
| Result | Value |
|---|---|
| Taper per foot | 1 in |
| Half angle (per side) | 2.3859 ° |
| Included angle | 4.7719 ° |
| Taper ratio (1 in this length) | 12 :1 |
Assumptions and limits
- A straight (conical) taper. A tapered part with a curved profile needs different math.
- Diameters measured perpendicular to the axis, at the two ends of the length given.
- Tolerance and standard-taper dimensions are not evaluated.
Common questions
What is a 1 inch per foot taper in degrees?
About 4.77° included, or 2.39° per side.
How do I set a lathe for a taper?
Set the compound to the half angle, or offset the tailstock by half the taper per foot times the work length in feet.
What is the difference between included angle and half angle?
The included angle is the full angle between the two sides; the half angle is the angle of one side to the axis.
What is a taper ratio?
The length over which the diameter changes by one unit, such as 12:1.
Sources
- Taper geometry from Machinery’s Handbook: taper per foot = (D − d) ÷ L × 12; tan(half angle) = (D − d) ÷ (2L).
Updated 2026-09-30