Taper Calculator - TPI, TPF, and Taper Angle

A taper calculator is a machining and metalworking tool that computes the geometric parameters of a tapered workpiece from a small set of measurements. Tapers appear whenever a cylindrical part changes diameter gradually, like a drill bit shank, a wooden dowel, a fence post, or a handrail transition. Enter any three of the major diameter, minor diameter, taper length, or included angle, and the calculator returns the rest plus the taper per inch, taper per foot, and the half-angle from the centerline.

• • Lathe and Shaper Setup: Machinists convert two end diameters and a length into a single taper per inch figure the compound rest can track.
• • Drill Bit and Chuck Sizing: Woodworkers and metalworkers confirm the taper per foot of Jacobs or Morse chuck tapers.
• • Handrail and Newel Post Fabrication: Trim carpenters size the transition between a larger newel post and a smaller handrail by matching the slope to the run.
• • Shaft and Pin Draft Angles: Pattern makers apply a small taper to wooden patterns so the casting can be drawn from the sand mold without damage.

Tapers are described in different ways depending on the trade. Some drawings state the major and minor diameters and the distance between them. Others state only the included angle and a single diameter, which is twice the half-angle that this calculator uses as its theta value. Self-holding tapers such as Morse and Jacobs are catalogued by a fixed taper per foot.

If you are laying out the joint where the taper meets another structural member, you can also plan the matching angle cut with our angle cut calculator to keep the miter geometry consistent across the project.

The taper calculation is built on a single linear ratio: the diameter drop across the tapered section divided by its length, with the half-angle as the inverse tangent of half that ratio.

• D_l: Major diameter at the larger end of the tapered section, in the selected unit.
• D_s: Minor diameter at the smaller end of the tapered section, in the selected unit.
• T_l: Taper length, the axial distance between the major and minor sections, not counting any straight portion.
• theta: Taper half-angle, the angle from the centerline of the workpiece to the sloped surface, in degrees. The included angle is 2 x theta.
• TPI: Taper per inch, the diameter change for each inch of axial length. A higher value means a steeper taper.

The two input modes are mirror images of the same relationship. In Standard mode you provide D_l, D_s, and T_l and the calculator returns TPI, TPF, theta, and the included angle. In By Half-Angle mode you provide D_l, D_s, and theta and the calculator returns TPI, TPF, the included angle, and T_l.

Omni Calculator's taper page defines taper per inch as the major diameter minus the minor diameter divided by the length, and the half-angle from the centerline as the inverse tangent of half that taper per inch.

For related layout work, a countersink depth calculator shares the same diameter-difference approach, and a bolt circle calculator carries the diameter math to a flat pattern of bolt positions.

Four terms come up in every taper drawing, handbook entry, and tooling chart. Understanding them is the difference between buying the right arbor and returning it the next day.

Self-holding tapers such as Morse and Jacobs typically sit at 0.5 to 0.75 inches per foot, while quick-releasing tapers run steeper and can exceed 2 inches per foot. Knowing the category up front prevents buying a Jacobs taper for a job that needs a precision Brown and Sharpe socket.

If you need to convert the diameter along the taper for a layout or a cut, this calculator works hand in hand with our miter angle calculator to carry the same trig into a compound or single miter setup, and our pitch diameter calculator covers threaded shafts with the same precision concerns.

Switching to the correct mode and entering three measurements is all it takes. The remaining values update as you type, and the unit selector handles inch and millimeter inputs.

1. 1 Choose Standard or By Half-Angle: Pick Standard if you know the diameters and length. Pick By Half-Angle if you know the diameters and the half-angle from the centerline and need the length.
2. 2 Select Inch or Millimeter: Pick inches or millimeters. Taper per inch always reports per inch, but the diameter and length use the unit you selected.
3. 3 Enter the Two Diameters: Type the major and minor diameters. Make sure D_l is the larger of the two or the calculator returns zeros to flag the inconsistency.
4. 4 Enter the Third Measurement: In Standard mode, type the taper length. In By Half-Angle mode, type the half-angle (theta) from the centerline in degrees; if your drawing gives the included angle, divide it by 2 first.
5. 5 Read the Outputs: Watch the taper per inch, taper per foot, half-angle, taper length, and included angle cards update as you type.

A woodworker is turning a newel post to a handrail with major end 3.0 in, minor end 1.5 in, and included angle 28 degrees. They divide by 2 to get 14 degrees, select By Half-Angle, and enter 3.0, 1.5, and 14. The calculator returns TPI = 0.249 in/in, TPF = 2.984 in/ft, taper length = 6.024 in, and included angle = 28 degrees.

A single calculator that covers the common taper notations removes round trips to the machinist handbook and protects the first cut.

• • Replaces Manual Trigonometry: Removes the need to convert between inverse tangent formulas, diameter ratios, included angles, and feet-to-inch scaling.
• • Prevents Mis-Matched Tooling: Confirms the taper per foot of a candidate arbor before ordering, so a Morse 2 is not paired with a Morse 3 socket by accident.
• • Speeds Lathe Setup: Translates three part dimensions into a single compound-rest half-angle the operator can dial in directly.
• • Supports Inch and Millimeter Workflows: Lets a workshop that buys from both domestic and metric catalogs enter values in either unit.
• • Documents the Decision: Records the taper per inch, half-angle, and full included angle for a setup so the same numbers can be reused for the next part.

On a busy shop floor the speed of switching between taper descriptions matters as much as the accuracy. A calculator that accepts diameters and length, or diameters and a half-angle from a drawing, removes the most common arithmetical mistakes on a setup sheet.

For related layout work, a countersink depth calculator shares the same diameter-difference approach for countersunk holes, and a bolt circle calculator carries the diameter math to a flat pattern of bolt positions. A taper calculator that covers both notations removes a step from the workflow.

Most taper calculations are exact for an ideal cone, but real-world variables push the actual part away from the math. Knowing them helps when a finished part does not match the drawing.

• • The calculator assumes a perfect cone. Real parts can have small flats, bell-mouth, or barrel profiles that the linear taper formula cannot represent.
• • It treats taper as a straight line from major to minor diameter. Complex multi-angle tapers such as a Jarno with a throat section need separate calculations per segment.

The first part off a new setup is the best time to verify the math against the real diameter with a micrometer at each end. Any systematic offset usually traces back to how the tool was set.

Wikipedia (Machine taper) groups machining tapers into self-holding and self-releasing (or steep) classes, with the Morse, Brown and Sharpe, Jarno, and Jacobs families identified by a taper per foot ratio that decides whether friction or a latch holds the tool.

Wikipedia (Morse taper) notes that the Morse taper is one of the most widely used families for drill chucks, lathe tailstocks, and reamers, with a per-family half-angle near 1.49 degrees from the centerline.

When you also need to verify the pitch diameter of a threaded shaft that sits on the tapered section, our pitch diameter calculator covers metric and unified thread dimensions for the same part.