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

The Taper Per Foot Formula

Taper per foot (TPF) is the standard way to specify tapers in the United States. It describes how much the diameter changes over a 12-inch length. A taper of 0.600 inches per foot means the diameter increases by 0.600 inches for every 12 inches of length. The formula is straightforward:

TPF = (D - d) × 12 / L, where D is the large diameter, d is the small diameter, and L is the taper length (all in inches). This gives the change in diameter per foot of length.

Taper per inch (TPI) is simply TPF divided by 12: TPI = (D - d) / L. Some drawings and references use TPI instead of TPF. Both describe the same geometric relationship, just scaled differently.

The included angle of the taper is related to TPF by trigonometry: half-angle = arctan(TPF / 24). The "24" comes from the fact that TPF is a diameter change, so you divide by 2 to get the radius change, and then by 12 to get radius change per inch. The included angle is twice the half-angle.

For example, a standard Morse #2 taper has TPF = 0.5994. Half-angle = arctan(0.5994 / 24) = arctan(0.02498) = 1.4307 degrees. Included angle = 2.8614 degrees. In degrees-minutes-seconds, that's 2°51'41".

The important thing to understand is that TPF is a ratio, not an angle. Two tapers with different diameters can have the same TPF, meaning they have the same angle. A 1-inch diameter taper with TPF 0.600 and a 6-inch diameter taper with TPF 0.600 have exactly the same angle — they're just different sizes.

Sine Bar Setup for Taper Inspection

A sine bar is a hardened steel bar with precision ground rolls at each end, separated by an exact center-to-center distance (typically 5 inches or 10 inches). By placing gage blocks under one end, you create a precise angle. The formula is: gage block height = sine bar length × sin(angle).

For taper inspection, you place the tapered part on the sine bar, set the sine bar to the nominal half-angle of the taper, and then run an indicator along the top surface. If the taper is correct, the indicator reading is constant along the entire length. If it varies, the taper angle is wrong — the direction and magnitude of the variation tell you how much the actual angle differs from the setup angle.

Setting up: Convert the taper specification to a half-angle in decimal degrees. Calculate the gage block stack: height = sin(half-angle) × sine bar length. Build the gage block stack to that height and place it under the high end of the sine bar. The low end sits on the surface plate. Place the tapered workpiece on the sine bar with the large end toward the gage blocks.

Reading the results: Zero the indicator at one end of the taper. Traverse to the other end. If the indicator reads zero at both ends and everywhere in between, the taper is correct. If the indicator shows a positive slope (reading increases toward the gage block end), the actual taper is steeper than the setting. If it shows a negative slope, the taper is shallower.

For a 5-inch sine bar, the gage block calculation for a Morse #2 taper (half-angle 1.4307°): height = sin(1.4307) × 5.000 = 0.02497 × 5.000 = 0.1249 inches. Build a gage block stack of 0.1249 inches.

Sine plates work the same way but provide a flat surface for holding parts. They're used for grinding and milling operations where the part needs to be held at a precise angle.

Morse Taper Reference

Morse tapers are the most common taper in the machine shop. They're used in drill press spindles, lathe tailstocks, milling machine drawbars, and countless toolholders. There are seven standard Morse taper sizes, numbered 0 through 6 (plus some rarely used larger sizes).

Morse Taper #0: Small end diameter 0.2520", large end diameter 0.3561", length 2.00", TPF 0.6246".

Morse Taper #1: Small end 0.3690", large end 0.4750", length 2.13", TPF 0.5986".

Morse Taper #2: Small end 0.5720", large end 0.7000", length 2.56", TPF 0.5994". This is the most common Morse taper in the shop — it fits most drill presses and small to medium lathes.

Morse Taper #3: Small end 0.7780", large end 0.9380", length 3.19", TPF 0.6024".

Morse Taper #4: Small end 1.0200", large end 1.2310", length 4.06", TPF 0.6233".

Morse Taper #5: Small end 1.4750", large end 1.7480", length 5.19", TPF 0.6315".

Morse Taper #6: Small end 2.1160", large end 2.4940", length 7.25", TPF 0.6257".

Notice that the TPF is not exactly the same across all Morse taper sizes — it varies from about 0.5986 to 0.6315. This is a historical artifact of the original Morse Manufacturing Company's design. A Morse #2 taper will NOT fit a Morse #3 socket, even though the TPFs are close. Each size is unique.

The self-holding property of Morse tapers comes from their shallow angle. The friction coefficient of steel-on-steel is about 0.15, and the half-angle of a Morse taper (about 1.4-1.5 degrees) is well below the friction angle. This means the taper wedges in and holds by friction alone. Ejection requires a drift key or a drawbar.

Brown & Sharpe and Other Standard Tapers

Brown & Sharpe tapers: Used primarily in milling machine spindles and some collet systems. B&S tapers have a TPF of 0.5000 for all sizes except #10, which is 0.5161. The consistent 0.500 TPF makes them slightly easier to calculate than Morse tapers. Common sizes are B&S #7 (used in small vertical mills), B&S #9 (common in horizontal mills), and B&S #11 and #12 (large horizontal mills).

R8 taper: Used in Bridgeport-style vertical milling machines. The R8 has a TPF of 3.500" — much steeper than Morse or B&S. This steep taper means R8 toolholders do NOT self-hold by friction. They require a drawbar to hold them in the spindle. The steep taper allows quick tool changes because there's no wedging action to overcome.

CAT (V-flange) tapers: Used in CNC machining centers. CAT 40 and CAT 50 are the most common. These use a 3.500 TPF (same as the R8 concept) and rely on the drawbar/retention knob for holding. The V-flange provides radial alignment and the drawbar provides axial clamping.

Jacobs taper: Used on drill chuck arbors. JT33 (the most common) has a TPF of 0.6240. JT1, JT2, JT3, and JT6 are also encountered. Jacobs tapers are small-diameter, self-holding tapers similar in concept to Morse tapers.

HSK (Hollow Shank Taper): The European standard for high-speed CNC machines. HSK is not a traditional taper in the same sense — it uses a 1:10 taper ratio (equivalent to about 1.146 TPF) with face contact for high-speed rigidity. HSK holders are lighter than CAT holders and provide better balance at high RPM.

The Tailstock Offset Method for Turning Tapers

On a manual lathe without a taper attachment, the most common way to turn a taper is the tailstock offset method. You shift the tailstock off-center so that the workpiece axis is no longer parallel to the carriage travel. As the carriage feeds along its normal path, the tool cuts a taper because the workpiece is angled.

The formula for tailstock offset is: Offset = (TPF × L_total) / 24, where L_total is the total length of the workpiece between centers (not just the length of the tapered section) and TPF is in inches per foot. Alternatively: Offset = (D - d) × L_total / (2 × L_taper), where D and d are the large and small diameters, L_total is the full workpiece length, and L_taper is the length of the tapered portion.

If the taper runs the full length of the workpiece (L_total = L_taper), the formula simplifies to: Offset = (D - d) / 2. This is the simplest case and is used for straight tapers that cover the entire piece.

Setting the offset: Measure the current tailstock center position by putting a dead center in the headstock and a dead center in the tailstock, bringing them together, and checking alignment with a dial indicator on the tailstock center. Zero the indicator. Then shift the tailstock by the calculated offset amount using the tailstock adjustment screws and the indicator.

Limitations: The tailstock offset method only works for workpieces held between centers. It cannot be used for chuck-mounted work. The offset changes the relative position of the centers, which means the live center does not seat fully in the center hole — this causes uneven wear on the center and limits the maximum offset to about 0.5 inches for most lathes. For steep tapers, use a taper attachment or a compound rest.

Compound rest method: For short tapers and steep angles, set the compound rest to the half-angle of the taper and feed the tool with the compound handwheel. This method is limited by the length of the compound slide (typically 3-4 inches of travel) but can produce any angle up to 60 degrees.

Angle Conversions: DMS, Decimal Degrees, and Radians

Machinists encounter angles in three formats, and converting between them is a daily task. Drawings may specify degrees-minutes-seconds (DMS), CNC programs use decimal degrees, and trig functions on scientific calculators may use any of the three (check your calculator mode!).

Degrees-Minutes-Seconds (DMS): 1 degree = 60 minutes (60'), 1 minute = 60 seconds (60"). So 2°51'41" means 2 degrees plus 51 minutes plus 41 seconds. To convert to decimal degrees: add degrees + (minutes/60) + (seconds/3600). So 2°51'41" = 2 + 51/60 + 41/3600 = 2 + 0.8500 + 0.01139 = 2.861 degrees.

Decimal degrees to DMS: Take the whole number part as degrees (2). Multiply the decimal part by 60 to get minutes: 0.861 × 60 = 51.67 minutes. Take the whole number as minutes (51). Multiply the decimal part by 60 to get seconds: 0.67 × 60 = 40.0 seconds. Result: 2°51'40".

Degrees to radians: Multiply by π/180. So 2.861° = 2.861 × 0.01745 = 0.04994 radians. Radians are used in most programming languages and some CNC functions.

Radians to degrees: Multiply by 180/π. So 0.04994 radians = 0.04994 × 57.2958 = 2.861 degrees.

The most common conversion mistake in the shop is forgetting that a calculator is in radian mode when you expect degree mode (or vice versa). If sin(30) gives you -0.988 instead of 0.500, your calculator is in radian mode. Switch to degree mode. This mistake has caused more scrap parts than most machinists want to admit.

Practical Tips for Taper Work

Bluing compound is your friend. When fitting a taper to a socket, apply Prussian blue or Dykem Hi-Spot to the taper, insert it into the socket, rotate it about 30 degrees, and remove it. The blue transfer pattern shows you where contact is occurring. A correct taper fit shows uniform contact over at least 75% of the surface, concentrated in the middle third of the taper length. Contact at the ends only means the taper angle is too steep. Contact in the center only means the taper angle is too shallow.

Taper reamers save time. For standard tapers (Morse, B&S, R8), a finish taper reamer produces a precise taper much faster than boring. Rough bore to within 0.010 to 0.015 of the final size, then finish with the reamer. Taper reamers are expensive ($50-$200) but they pay for themselves on the second job.

Check taper with a taper gauge. Taper plug gauges and taper ring gauges have scribe lines that indicate acceptable positioning. When the taper seats in the gauge, the end of the part should fall between the two scribe lines. Above the upper line means the part is undersized; below the lower line means oversized.

Lapping for precision. For tapers that must seal (valve seats, spindle tapers), final fitting is done by lapping. Apply lapping compound to the male taper, insert it into the female socket, and rotate it back and forth while applying light axial pressure. The compound removes high spots and produces a matched, sealing fit. Clean thoroughly after lapping — lapping compound left in a spindle taper will destroy the seating surface over time.

Don't forget thermal expansion. A steel taper that fits perfectly at room temperature may loosen or tighten at operating temperature. Machine tool spindles run warm (typically 5-15°C above ambient), which slightly expands the bore. This is by design — the taper holder is clamped cold and the thermal expansion ensures a tighter fit at operating temperature.