How to use the taper calculator
- Choose what you know: both diameters, a half angle, or a taper per foot.
- Enter the large diameter and the length of the taper in inches or millimeters, plus the small diameter, angle or rate.
- To set up a lathe between centers, enter the overall workpiece length and read the tailstock offset.
- Read the taper per unit, per foot, the ratio, the half and included angles and the small diameter.
Formula
Taper per unit length = (large - small) / length
Half angle = arctan((large - small) / (2 x length)); included angle = 2 x half angle
Small diameter from a half angle = large - 2 x length x tan(half angle)
Tailstock offset = (large - small) / 2 x workpiece length / taper length
Taper per foot is the taper per unit length times 12. The ratio is the length that goes with a change of one unit in diameter.
A 6 inch taper from 2 inches to 1.5 inches
Diameters of 2 and 1.5 inches over 6 inches give a taper of 0.08333 per inch, 1 inch per foot and a ratio of 1:12. The half angle is 2.38594 degrees and the included angle is 4.77189 degrees. Turned on a 10 inch workpiece between centers, the tailstock offset is 0.41667 inch.
In metric, a taper from 50 mm to 40 mm over 200 mm is 0.05 per millimeter, a half angle of 1.4321 degrees and 0.6 mm of diameter change for every 12 mm of length. A half angle of 3 degrees on a 2 inch diameter over 6 inches ends at a small diameter of 1.37111 inches.
Ways to state a taper
Tapers are given as a ratio, as a change in diameter per foot, as a half angle on each side of the center line, or as an included angle between the two sides. Machine and tool shank tapers are often stated per foot or as a ratio, while angles are common in drawings. The page converts between them so that you can set a compound slide or a tailstock from whichever one the drawing gives.
For related shop geometry see the hole circle calculator, the cone flat pattern calculator, the miter angle calculator and the slope calculator.
Tailstock offset
Offsetting the tailstock tilts the work between centers so that a straight cut makes a taper. The offset depends on the length of the whole piece as well as the taper itself, which is why the page asks for both. It is an approximate method: the center holes wear at an angle, so make a trial cut, measure it and adjust.
Assumptions and limits
- The taper is a straight cone with the large end first, and the small diameter must be less than the large one.
- Lengths and diameters share one unit, inches or millimeters.
- The tailstock offset is the textbook estimate for turning between centers.
- Standard tapers such as Morse and Jacobs have set sizes that you should look up rather than compute.
Frequently asked questions
How do I calculate taper per foot?
Subtract the small diameter from the large one, divide by the length in inches and multiply by 12. A change of 0.5 inch over 6 inches is 1 inch per foot.
How do I convert a taper to an angle?
The half angle is the arctangent of the diameter change divided by twice the length. For 0.5 over 6 that is 2.386 degrees, so the included angle is 4.772 degrees.
What is the half angle?
It is the angle between one side of the taper and the center line. The included angle is the angle between the two sides, twice as large.
How do I find the tailstock offset?
Take half of the diameter change, multiply by the full workpiece length and divide by the length of the taper. The page also computes it for you.