Taper Calculator

What this calculator does

Turn a taper's diameters and length into the numbers you need to cut it on a lathe. A taper is a cone-shaped section on a shaft or inside a hole. It's used wherever two parts need to lock together and line themselves up on center. Examples include drill chucks, lathe centers, tool holders, propeller shafts, pins and handles. A drawing can describe a taper in several ways. It might give two diameters and a length, a taper per foot, a ratio such as 1:20, or an angle. To set up a lathe to cut it, you need the angle or an offset.

This calculator converts between them. Give it both diameters and the length, or the large diameter and the included (full) angle. It returns the taper per inch (or millimeter), the taper per foot, the included and half angles, and the taper ratio. It also gives the tailstock offset for turning the taper between centers. The half angle is the setting for a compound rest, measured from the lathe's centerline (axis).

Where it's used

Why this math matters outside the classroom:

• Lathe setup. A machinist sets the compound rest to 1.7899° to cut a 3/4 in per foot taper on a shaft.
• Reading drawings. An apprentice turns a 1:20 taper on a 30 mm shaft into a 2.8642° included angle for the setup sheet.
• Tailstock offset. A turner offsets the tailstock 0.1875 in to cut a 1 in to 0.875 in taper over 2 in on a 6 in part.
• Woodturning. A woodturner sizes a chair leg that tapers from 1.5 in to 1 in over 15 in, a 0.4 in per foot taper.
• Inspection. An inspector measures 30 mm and 25 mm diameters 100 mm apart to confirm a 1:20 taper.

Grade level and standards: Grades 10–11; Common Core State Standards for Mathematics HSG.SRT.C.8. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

1. Choose whether you know both diameters, or the large diameter and the included angle.
2. Pick inches or millimeters. All lengths use the same unit.
3. Enter the diameters (or the angle) and the length of the taper along its centerline.
4. If you plan to use the tailstock offset method, enter the overall length between centers. Then read the results.

What the inputs mean

Included angle

The included angle is the full angle of the cone. Many drawings and lathe dials use the half angle instead, which is shown in the results.

Length of the taper (L)

Measure the taper length along the centerline of the part, not along the sloping surface.

Overall length of the workpiece

The tailstock offset method tilts the whole workpiece. So the offset (setover) depends on the total length between centers, not just the tapered section. If the entire part is tapered, enter the same value as the taper length.

Formula and method

taper per inch = (D − d) ÷ L
taper per foot = 12 × (D − d) ÷ L      (inches)
half angle = arctan((D − d) ÷ (2L))
included angle = 2 × half angle
taper ratio = 1 : L ÷ (D − d)
tailstock offset = (D − d) ÷ 2 × (overall length ÷ L)

In the formulas, D is the large end diameter, d is the small end diameter, and L is the length of the taper along its centerline. Arctan turns a slope (rise over run) into an angle. The overall length is the full length of the workpiece between centers.

Picture a taper sliced down its centerline. Each side forms a right triangle with the centerline. The rise is half the diameter change, (D − d) ÷ 2, and the run is the length L. The tangent of the half angle is rise over run, so the half angle is arctan((D − d) ÷ 2L).

For the tailstock offset method, you tilt the work by moving the tailstock sideways. Over the taper length, the radius (half the diameter) must change by (D − d) ÷ 2. So over the full length between centers, the offset is that amount scaled up by overall length ÷ L.

Worked example

A shaft tapers from 1 in to 0.875 in over 2 in, on a 6 in long workpiece turned between centers.

1. Taper per inch: (1 − 0.875) ÷ 2 = 0.0625 in/in.
2. Half angle: arctan(0.125 ÷ 4) = 1.7899°.
3. Included angle: 2 × 1.7899 = 3.5798°.
4. Taper per foot: 0.0625 × 12 = 0.75 in/ft.
5. Tailstock offset: 0.125 ÷ 2 × (6 ÷ 2) = 0.1875 in.

Result: The taper is 3/4 in per foot, an included angle of 3.5798° (1.7899° per side), and a 1:16 ratio. It needs a 0.1875 in tailstock offset.

Understanding the result

Taper per foot is the traditional way to describe inch tapers. For example, 3/4 in per foot is about 3°35′ included. Metric drawings more often give a ratio such as 1:20 or 1:50. That means 1 mm of diameter change for every 20 or 50 mm of length.

Set a compound rest to the half angle, measured from the lathe's centerline. If your compound's dial reads from the cross-slide instead, set it to 90° minus the half angle.

Assumptions and limitations

• A straight (cone-shaped) taper that is round all along its length.
• The tailstock offset formula assumes the work is held between centers and the taper lies within that length.
• The tailstock offset method suits small angles. At larger offsets, the centers sit crooked in their holes. So steep tapers are cut with the compound rest or a taper attachment.
• Standard taper systems, such as machine tool tapers and pipe threads, have their own published dimensions. Use those when matching an existing standard.
• It doesn't calculate the gauge-line position or the length along the sloping side.

Frequently asked questions

How this calculator is checked

Method: Right-triangle trigonometry on the taper's half section: tan(half angle) = (D − d)/(2L); taper per foot = 12(D − d)/L; tailstock setover = (D − d)/2 × (overall length / taper length). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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