Taper Calculator

Taper Calculator for taper per inch, per foot or per meter, half and included angles and the 1:X ratio from two end diameters and a length, or the missing diameter or taper length.

How a Taper Calculator Works Out a Taper

A taper is a steady change in diameter along a length, like a cone with its point cut off. Machine tapers lock tools in spindles, while pins and pipe threads use tapers to align or seal parts.

A taper calculator measures that change from three numbers: the large end diameter, the small end diameter and the length between them.

To calculate a taper, subtract the small end from the large end, then divide by the length. The answer is the taper per inch, and every other form of the taper comes from it.

$$TPI=\frac{D-d}{L}$$

D is the large end diameter, d is the small end diameter and L is the length of the tapered section only, not the whole part. All three must be in the same unit.

A digital taper calculator runs this one division, then converts the result into taper per foot, a ratio and two angles. Working backward from any of those forms gives a missing diameter or length.

Worked Example: 1.25 in to 1 in Over 3.5 in

A shaft drops from 1.25 in to 1.000 in over 3.5 in. The taper per inch is 0.25 ÷ 3.5 = 0.07143 in/in, which is 0.857 in/ft and a ratio of 1:14 on diameter.

The half angle is arctan(0.07143 ÷ 2) = 2.0454°, or 2° 2′ 43″, and the included angle is 4.0908°. Carried past the small end, the same taper would come to a point 14 in further on.

Taper per Foot and Taper per Inch

Taper per foot (TPF) is the change in diameter over 12 in of length. Most US drawings and tool catalogs quote tapers this way, so a taper calculator usually reports it first.

$$TPF=\frac{12\times (D-d)}{L}$$

To find taper per inch, divide the diameter change by the length in inches, or divide the taper per foot by 12. A Morse #3 at 0.60235 in/ft works out to 0.05020 in/in.

To estimate taper per foot from a part you can measure, mic both ends of a known length and scale up to 12 in. If 1.000 in shrinks to 0.950 in over 2 in, the TPF is 12 × 0.050 ÷ 2 = 0.300 in/ft.

Taper is always measured on the diameter in these forms, not on one side. A part that loses 0.025 in per side over an inch has a taper of 0.050 in/in, because both sides move in.

Taper Calculator in mm

A taper calculator in mm uses millimeters for both diameters and length, so the taper comes out in mm per mm. Multiply by 1,000 for taper per meter, which reads cleanly on metric drawings.

A part that goes from 30 mm to 25 mm over 100 mm has a taper of 0.05 mm/mm, or 50 mm/m. That is a 1:20 taper with a 1.4321° half angle, the same rate as the Jarno and metric tool tapers.

How to Find the Taper Angle in Degrees

To find the taper angle, take the arctangent of half the taper per inch. The result is the half angle, measured from the center line to one side of the cone.

$$\alpha =\arctan \left(\frac{D-d}{2L}\right)$$

α is the half angle, also called the angle per side or compound angle. The included angle is 2α, the full angle between opposite sides of the cone. Drawings use both, so check which one the print gives.

The half angle is the one you set on a lathe compound rest, because the tool moves along one side only. The Countersink Depth Calculator works the other way, from the included angles of 82°, 90° and 100° countersinks.

Use a taper calculator in degrees and minutes when your compound dial is graduated that way. Most dials read to the degree, so 2° 2′ 43″ sets as just over 2°, then gets trimmed by trial cuts.

Taper per Foot to Degrees

The chart converts common taper per foot values to angles with the same formulas a taper calculator uses. The 0.625 in/ft row is the old rule of thumb for Morse tapers, which actually run from 0.599 to 0.632 in/ft.

Going the other way, taper per foot equals 24 × tan(α). A 3° half angle is 24 × tan 3° = 1.258 in/ft, and a 10° half angle is 4.232 in/ft.

What a Taper Ratio Means

A taper ratio such as 1:20 means the diameter changes by 1 unit over 20 units of length. It works in any unit, which is why metric and international standards prefer it.

$$Ratio=1:\frac{L}{D-d}$$

The rate of taper is the same idea written as a single number. A taper ratio calculator divides the length by the diameter change, so 0.25 in over 3.5 in gives 1:14.

Watch for ratios given per side. Some older drawings write "1 in 20 per side," which is 1:10 on diameter. A taper calculator that takes the ratio on diameter needs that value halved first.

1:10 Taper Calculation

A 1:10 taper drops 1 mm in diameter for every 10 mm of length. That is 0.1 mm/mm, 100 mm/m or 1.2 in/ft on the inch side.

Its half angle is arctan(0.05) = 2.8624°, or 2° 51′ 45″, with a 5.7248° included angle. Over a 40 mm long taper, the diameter changes by exactly 4 mm.

Taper Length and the Missing Diameter

A taper length calculator rearranges the same formula. Divide the diameter change by the taper per inch to get the length, or multiply the taper per inch by the length to get the change.

$$L=\frac{D-d}{TPI}d=D-TPI\times L$$

Use the length form when both diameters are fixed and the rate is set by a standard. A 1:20 taper from 20 mm down to 16 mm needs (20 - 16) × 20 = 80 mm of length.

Use the diameter form when you know one end. A shaft taper calculator setup with a 1.500 in large end, a 0.750 in/ft rate and 2 in of length gives a small end of 1.500 - 0.0625 × 2 = 1.375 in.

A steep rate over a long length can run past the cone point. When the small end comes out at zero or below, the taper is longer than the cone it describes, so shorten the length or check the rate.

Standard Machine Taper Chart

Machine tapers hold tools by friction or a drawbar, so the rate must match exactly. Mic an unknown shank at two points, run the numbers through a tool taper calculator and match the rate below.

Morse rates come from Machinery's Handbook Pocket Companion standard tapers, which shows each size has its own rate. Jacobs and Brown & Sharpe values use the end diameters and lengths published by Newman Tools.

Sources differ on Morse #4½. The machine taper tables on Wikipedia give 1:19.128, while some metric data sheets list 1:19.231. Measure a #4½ shank before relying on either rate.

Brown & Sharpe tapers run 0.500 in/ft except #10. Newman's #10 data, 1.28927 in to 1.04465 in over 5-11/16 in, works out to 0.51612 in/ft, so a #10 shank reads about 3% off the usual rate.

Morse, Jarno and Brown & Sharpe are self-holding tapers. Their small angles, about 2.4° to 3° included, let friction alone hold the tool. Jacobs chuck tapers vary more, from about 2.8° included on #0 up to 4.7° on #2 and #2 Short.

The 7:24 CAT, BT and NMTB tapers release freely, so they need a drawbar. The R8 collet taper, at 16° 51′ included, also relies on a drawbar to pull the collet closed.

NPT pipe threads use a 1:16 taper on diameter, or 0.750 in/ft. The Thread Calculator and Pitch Diameter Calculator cover the thread form that rides on that taper.

Turning a Taper on a Lathe

A lathe can cut a taper four ways. The taper calculator numbers feed each one differently, and the right method depends on taper length, steepness and whether the taper is inside or outside.

Compound Rest Setting

Take the half angle from the taper calculator and set the compound to it from the lathe axis, not to the included angle. For the 1.25 in to 1 in example, that is 2° 2′ 43″, and the compound travel along the taper is 3.5 ÷ cos(2.0454°) = 3.5022 in.

Many compound dials read 0° when the slide sits square to the axis. On those, set 90° - α instead, which is 87.9546° for the example. Check your dial's zero before cutting.

Tailstock Offset

The tailstock offset method moves the tailstock center sideways so the whole part between centers sits at the taper angle. Give the taper calculator the full length between centers, not just the tapered length.

$$Offset=\frac{TPF\times L_{part}}{24}$$

TPF is the taper per foot and L_part is the length between centers in inches. An 8 in part with a 0.857 in/ft taper needs 0.857 × 8 ÷ 24 = 0.2857 in of offset.

A member of the Hobby-Machinist forum works an MT3 example the same way: 4.220 in × 0.6024 ÷ 24 gives about 0.106 in. The same thread notes that a tool set 1/8 in off center height turns the taper into a curve.

Speed for taper turning follows the smaller diameter when you work close to the spindle limit. The Spindle Speed Calculator finds RPM at both ends of the cut.

Checking a Taper With a Sine Bar

A sine bar sets the angle from the taper calculator with gauge blocks under one roller. The stack height is the bar length times the sine of the angle you want level.

$$H=S\times \sin (\alpha )$$

H is the gauge block stack and S is the sine bar length between rollers. For a part held on centers on the bar, use the half angle: a 5 in bar needs 5 × sin(2.0454°) = 0.1785 in.

For a part lying on its tapered side on the bar, the top edge sits at the included angle, so use 2α. The same 5 in bar then needs 0.3567 in. A dial indicator run along the top edge should read flat once the angle is right.

A quick size check works too. On a straight taper, the diameter halfway along equals the average of the two ends, so the example should mic 1.125 in at 1.75 in from either end.

Taper Calculator Answers for Odd Cases

Some taper jobs start with too little data or use the word taper for a different shape. The math still works if you know which angle or rate the job means.

How do I calculate a taper with only one diameter?

You need a second fact, either the other diameter or the rate. With a known standard, the taper calculator gives the missing end: a Morse #2 at 0.59941 in/ft over 3 in changes diameter by 0.1499 in.

What is the difference between taper and draft angle?

Draft is the small angle on molded and cast walls that lets a part leave the mold. It is quoted per side, like a half angle, so a 1° draft on a round core equals a 2° included taper.

Does a machining taper calculator work for road tapers?

No. Work zone lane tapers under the FHWA Manual on Uniform Traffic Control Devices (MUTCD) size their length from lane width and speed. A machining taper calculator measures diameter change, which is a different problem.