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Taper & Angle Calculator — Taper Per Foot, Sine Bar Heights & Standard Taper Reference

Calculate Taper from Diameters, Sine Bar Stack Heights, Tailstock Offset, and Reference Morse, Brown & Sharpe, and R8 Tapers

Free taper and angle calculator for machinists, toolmakers, and CNC programmers. Enter two diameters and the taper length to calculate taper per foot (TPF), taper per inch (TPI), included angle, and half angle in degrees-minutes-seconds. Includes sine bar height calculation for any angle and bar length, tailstock offset for turning tapers between centers, and a complete reference table of standard tapers including Morse (#0 through #6), Brown & Sharpe (#1 through #18), R8, Jarno, 7/24 (CAT/BT), and HSK. Outputs DMS angles and decimal degrees.

Pro Tip: When setting a taper on a lathe using the compound slide or taper attachment, always set it to the half angle, not the included angle. The included angle is the total angle of the taper measured across both sides. The half angle is what the cutting tool traces on one side. If a drawing calls out a 3-degree included angle taper, set your compound to 1.5 degrees. For sine bar work, double-check your calculation by measuring both ends of the part with a micrometer after grinding — calculated tapers should match measured diameters within 0.0002" or something is set up wrong.

Calculate lathe cutting parameters for taper turning

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Read the taper and angle machining guide

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Taper & Angle Calculator
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How It Works

  1. Enter Taper Dimensions

    Input the large diameter, small diameter, and taper length. The calculator determines the included angle, half angle, taper per foot, and taper per inch. If you know the angle and one diameter, use the reverse mode to calculate the second diameter.

  2. Select Standard Taper (Optional)

    Choose a standard taper from the reference table to auto-fill dimensions. Options include Morse Tapers #0-#6, Brown & Sharpe #1-#18, R8, Jarno, 7/24 (CAT/BT), and HSK. Each standard includes the exact dimensions, TPF, and included angle.

  3. Calculate Sine Bar Height

    Enter the desired angle and sine bar length (5" or 10" standard) to calculate the gauge block stack height needed to set that angle. Useful for surface grinding tapers, inspecting taper angles, and setting compound rest angles with precision.

  4. Calculate Tailstock Offset

    For turning tapers between centers on a lathe, enter the large diameter, small diameter, total part length, and taper length. The calculator computes the tailstock offset distance needed. This method is limited to slight tapers and long parts.

  5. Review Output

    See all taper parameters: TPF, TPI, included angle (DMS and decimal), half angle, sine bar height for 5" and 10" bars, and tailstock offset. Copy values directly to your setup sheet or program. DMS output matches the format used on compound rest dials.

Built For

  • Machinists setting compound slide angles for turning Morse tapers on manual lathes
  • Toolmakers calculating sine bar gauge block stacks for grinding taper gauges and mandrels
  • CNC programmers computing half angles for G-code taper turning and taper boring cycles
  • Machine repair technicians measuring worn spindle tapers and comparing to Morse or R8 specifications
  • Quality inspectors verifying taper angles with sine bars and dial indicators during incoming inspection
  • Engineering students learning the relationships between TPF, included angle, half angle, and taper ratio

Features & Capabilities

Taper Per Foot & Per Inch

Calculates taper per foot (TPF) and taper per inch (TPI) from any two diameters and a length. TPF is the most common taper specification in American shop practice. Results match published values in Machinery's Handbook.

DMS Angle Output

Displays included angle and half angle in degrees, minutes, and seconds (DMS) format, which matches the graduations on compound rest dials, protractors, and angle plates. Also shows decimal degrees for CNC programming.

Sine Bar Calculator

Computes gauge block stack height for any angle on 5", 10", or custom-length sine bars. Stack height = bar length x sine(angle). Essential for precision grinding, inspection, and setting angles to better than 1 minute of arc.

Tailstock Offset

Calculates the lateral tailstock offset distance for turning tapers between centers. Shows the offset direction and distance, plus a warning when the taper is too steep for this method (generally limited to tapers under 3 degrees half angle).

Standard Taper Reference

Complete reference table of Morse Tapers #0-#6, Brown & Sharpe #1-#18, R8, Jarno, 7/24 (CAT 40/50, BT 30/40/50), and HSK. Includes large end diameter, small end diameter, length, TPF, included angle, and tang/draw bar specifications.

Reverse Calculation

Enter any three of the four taper parameters (large diameter, small diameter, length, angle) and the calculator solves for the missing value. Useful when working from a drawing that specifies the angle and one diameter but not the other.

Frequently Asked Questions

The included angle is the total angle of the taper measured between the two sloped surfaces. The half angle is exactly half of that — the angle between one surface and the centerline. When setting a compound rest or taper attachment, you set the half angle because the tool cuts along one side of the taper while the part rotates around the centerline. A Morse #2 taper has an included angle of approximately 2 degrees 51 minutes, so the half angle is approximately 1 degree 25.5 minutes.
Place the sine bar on a surface plate. Stack gauge blocks under one end to the calculated height. Place the tapered part on the sine bar with the taper surface facing up. Run a dial indicator along the top surface. If the indicator reads zero change across the taper length, the angle is correct. Any deviation in the indicator reading tells you the taper angle is off. One thousandth of indicator reading per inch of travel equals approximately 0.001 TPI error.
Morse #2 (MT2) is by far the most common. It's used in the spindle of most drill presses, smaller milling machines, and the tailstock of many lathes. MT3 is common in larger machine tools, and MT4 and MT5 are found in large lathes and heavy boring mills. MT1 appears in smaller bench-top machines and portable drills. The R8 taper, while not a Morse taper, is found in most Bridgeport-style vertical milling machines.
Use the tailstock offset method for long, shallow tapers where the workpiece is held between centers. It allows the full length of the bed for the carriage travel. Use the compound slide for short, steep tapers where the taper length is within the compound's travel range (typically 3-4 inches). The taper attachment, if your lathe has one, is the most versatile method — it handles both long and moderate tapers while maintaining automatic carriage feed.
Taper per foot (TPF) is a dimensional specification that's easier to measure and verify in the shop than an angle. You can check TPF with a micrometer at two points along the taper and some basic arithmetic. Measuring a small angle accurately requires a sine bar or precision protractor. TPF also translates directly to machining setup: the tailstock offset distance equals TPF times length divided by 24. For these practical reasons, American shop practice has traditionally preferred TPF over angular specifications.
Disclaimer: Taper calculations are based on geometric formulas and published standard taper dimensions. Actual machined tapers are subject to machine tool accuracy, setup precision, and measurement uncertainty. Standard taper dimensions are from Machinery's Handbook and manufacturer specifications. Always verify critical tapers with gauges or precision measurement instruments. ToolGrit is not responsible for machining errors or taper fit problems resulting from calculator use.

Learn More

Machinist

Taper Calculations for Machinists: TPF, Sine Bars, Morse Tapers, and Tailstock Offsets

Complete reference for taper calculations including taper per foot, sine bar gage block stacks, Morse and Brown & Sharpe taper charts, tailstock offset formula, and DMS angle conversions.

Read Guide →

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