Lathe Taper Length: Calculate Z from Major and Minor X Diameters

Calculate lathe taper length from the major X diameter, minor X diameter and taper angle. The axial Z distance uses the radial change—half the diameter difference—and the angle specified on your drawing.

Open the free X-diameter and angle to Z calculator →

What to enter

• Major X: the larger full diameter.
• Minor X: the smaller full diameter.
• Angle: either the taper angle to the spindle axis or the full included angle.

Use inches for both diameters or millimetres for both. The answer uses the same length unit. These formulas assume a straight, concentric taper between the stated diameters.

The formula for lathe taper length

The cross-section forms a right triangle. Its rise is the radial change and its run is the axial length Z. For the angle θ measured to the spindle axis:

Radial change = (major X − minor X) / 2
Z = (major X − minor X) / (2 × tan θ)

Angles in this guide are degrees. If your calculator or spreadsheet expects radians, convert the degree value before evaluating the tangent.

Inch example: X2.000 to X1.000 at 30°

Suppose the major diameter is 2.000 in and the minor diameter is 1.000 in. The drawing specifies a 30° angle to the axis.

1. Diameter difference: 2.000 − 1.000 = 1.000 in.
2. Radial change: 1.000 / 2 = 0.500 in.
3. Z = 0.500 / tan(30°) = 0.866025 in, rounded to six decimal places.

This gives the distance along the axis. It is not the sloping surface length and does not determine a signed machine coordinate by itself.

What if 30° is the included angle?

The full included angle spans both sides of a symmetrical taper. Divide it by two before using the tangent formula:

θ = included angle / 2

For the same X2.000 and X1.000 diameters with a 30° included angle, θ is 15°. The calculation becomes 0.500 / tan(15°), giving Z = 1.866025 in. That is a different taper from the 30°-to-axis example. Check the drawing convention before choosing the calculator option.

Metric example using the same taper

Convert the inch example exactly: 2 in = 50.8 mm and 1 in = 25.4 mm. At 30° to the axis, the radial change is 12.7 mm. Z = 12.7 / tan(30°) = 21.997045 mm, rounded. This equals the unrounded inch result multiplied by 25.4. Converting units changes the length values, not the angle.

Z distance versus a programmed Z coordinate

Keep the calculated distance separate from your program’s origin and direction. If the first taper endpoint is at Z0 and the second endpoint lies in the negative Z direction, the second coordinate would be −0.866025 in for the first example. A different origin or direction changes that coordinate. Confirm your machine’s X diameter or radius convention, tool compensation and drawing datums before programming.

Check the result before using it

• Enter full diameters in the calculator, even if your control uses another convention.
• Use a major diameter greater than the minor diameter.
• Use a half-angle greater than 0° and less than 90°.
• Do not mix an included angle with a half-angle formula.
• Keep sufficient precision through the calculation and round to the drawing’s requirements at the end.

Related shop tools

Use the taper Z calculator to repeat these examples. If you know both diameters and length and need the angle instead, choose “Taper — two diameters + length” in the same tool. For spindle and feed arithmetic, use the turning calculator. Explore our CNMG, DNMG and VNMG guide when preparing the tooling selection.

Frequently asked questions

Do I divide the X diameter difference by two?

Yes, when X values represent diameters. The radial change is (major diameter − minor diameter) ÷ 2. Use that radial change with the taper half-angle to calculate axial Z distance.

Is the calculated Z length a signed machine coordinate?

No. It is a positive axial distance. The programmed endpoint also depends on the starting coordinate, direction of travel, work offset and the machine’s programming convention.

What if the drawing gives the included taper angle?

Divide the included angle by two before using Z = (D − d) ÷ (2 × tan θ), where θ is measured from the spindle axis. Keep both diameters in the same length unit.