Taper Calculator
Enter the large diameter, small diameter, and taper length to get the full set of taper outputs: taper per inch, taper per foot, taper ratio, half-angle, and included angle. Switch between metric and imperial units instantly. Use the lookup table below to match your workpiece against standard Morse, Jacobs, Brown and Sharpe, or Jarno taper sizes.
What is a taper?
A taper is a gradual reduction in the diameter of a cylindrical or conical surface along its length. Tapering is used throughout machining and engineering: lathe chucks and spindles rely on tapered seats to hold cutting tools securely, pipe fittings use tapered threads to form leak-tight joints, and structural pins use taper fits to resist shear while remaining removable. The geometry of a taper is described by a small set of values - the large and small diameters, the axial length, the resulting taper per unit length, and the cone angle.
Taper formulas: TPI, TPF, ratio, and angle
All taper quantities derive from three measured dimensions: the large diameter (D1), the small diameter (D2), and the taper length (L). The diameter difference delta D = D1 - D2 drives everything else. Taper per inch (TPI) = delta D / L. Taper per foot (TPF) = 12 x TPI. Taper ratio is expressed as 1:X where X = L / delta D, so a 1:20 taper changes diameter by 1 unit for every 20 units of length. The half-angle theta = arctan(delta D / (2 x L)) and the included angle (the full cone angle) = 2 x theta. For example, a workpiece with D1 = 1.231 in, D2 = 1.020 in, and L = 3.875 in gives delta D = 0.211 in, TPI = 0.0544, TPF = 0.653, ratio 1:18.4, half-angle 1.56 deg, and included angle 3.12 deg - exactly matching a #4 Morse taper.
Self-holding vs releasing tapers
Whether a taper locks in place by friction or releases freely depends almost entirely on the half-angle. Half-angles below about 3 degrees generate enough normal force at the contact surface that friction alone prevents the tool from sliding out under load. These are called self-holding or slow tapers. Morse tapers (roughly 1.47 deg to 1.59 deg half-angle), Brown and Sharpe tapers (about 1.19 deg), and Jarno tapers (1.43 deg) are all self-holding. Above about 3 degrees, the taper becomes self-releasing: the tool must be retained by a draw bar, bolt, or other mechanical fastener. Steep-taper tooling such as CAT 40, BT 40, and HSK systems uses half-angles well above this threshold so that a power drawbar can clamp and release quickly.
How to use this calculator
Enter the large diameter (D1), the small diameter (D2), and the taper length (L) in either inches or millimetres. All six taper parameters update instantly. If you are trying to identify a standard taper, compare your computed TPI and half-angle values against the reference table below - or measure the large diameter and length, look up the matching row, and confirm the small diameter. When working in metric, the "taper per foot" output becomes "taper per metre" (diameter change per 1000 mm), which is the analogous figure used in ISO taper specifications.
Standard taper reference
| Standard | Size | Large dia. (in) | Small dia. (in) | Length (in) | TPI (approx.) | Half-angle (deg) |
|---|---|---|---|---|---|---|
| Morse | MT0 | 0.3651 | 0.2520 | 1.938 | 0.0584 | 1.67 |
| Morse | MT1 | 0.4750 | 0.3690 | 2.063 | 0.0514 | 1.47 |
| Morse | MT2 | 0.7000 | 0.5720 | 2.500 | 0.0512 | 1.47 |
| Morse | MT3 | 0.9380 | 0.7780 | 3.063 | 0.0523 | 1.50 |
| Morse | MT4 | 1.2310 | 1.0200 | 3.875 | 0.0544 | 1.56 |
| Morse | MT5 | 1.7480 | 1.4750 | 4.938 | 0.0554 | 1.59 |
| Morse | MT6 | 2.4940 | 2.1160 | 7.000 | 0.0540 | 1.55 |
| Morse | MT7 | 3.2700 | 2.7500 | 9.500 | 0.0547 | 1.57 |
| Brown and Sharpe | B&S 1 | 0.2392 | 0.2000 | 0.938 | 0.0417 | 1.19 |
| Brown and Sharpe | B&S 5 | 0.5232 | 0.4500 | 1.750 | 0.0418 | 1.20 |
| Brown and Sharpe | B&S 9 | 1.0670 | 0.9001 | 4.000 | 0.0417 | 1.19 |
| Brown and Sharpe | B&S 12 | 1.7968 | 1.5001 | 7.125 | 0.0417 | 1.19 |
| Jacobs | JT0 | 0.2500 | 0.2284 | 0.438 | 0.0493 | 1.41 |
| Jacobs | JT1 | 0.3840 | 0.3334 | 0.656 | 0.0770 | 2.21 |
| Jacobs | JT2 | 0.5590 | 0.4876 | 0.875 | 0.0816 | 2.34 |
| Jacobs | JT3 | 0.8110 | 0.7461 | 1.219 | 0.0532 | 1.52 |
| Jacobs | JT6 | 0.6760 | 0.6241 | 1.000 | 0.0519 | 1.49 |
| Jarno | Jarno 4 | 0.5000 | 0.4000 | 2.000 | 0.0500 | 1.43 |
| Jarno | Jarno 8 | 1.0000 | 0.8000 | 4.000 | 0.0500 | 1.43 |
| Jarno | Jarno 12 | 1.5000 | 1.2000 | 6.000 | 0.0500 | 1.43 |
Approximate TPI and half-angle values for common machine tool tapers. Use these to identify or specify a standard taper for your application.
Frequently asked questions
What is taper per inch (TPI) and how is it calculated?
Taper per inch (TPI) is the reduction in diameter for every inch of axial length along a tapered surface. It is calculated as TPI = (D1 - D2) / L, where D1 is the large diameter, D2 is the small diameter, and L is the taper length, all measured in inches. TPI is the most common way to specify a taper for machining because it relates directly to the compound slide angle on a lathe.
What is the difference between a Morse taper and a Jacobs taper?
Morse tapers are used in lathe spindles, tailstocks, and drill press quills to hold tools such as drill bits, centers, and chucks. They have a shallow half-angle (about 1.47 to 1.59 degrees) that makes them self-holding. Jacobs tapers are specifically designed for the shanks of Jacobs drill chucks and similar accessories. They are similar in angle but shallower in some sizes and cover a different diameter range. Morse tapers come in sizes MT0 through MT7; Jacobs tapers come in sizes JT0 through JT6 plus JT33.
How do I convert TPI to taper angle?
The half-angle in degrees is theta = arctan(TPI / 2). The "divide by 2" accounts for the fact that the diameter change is split equally on each side of the centreline. For example, TPI = 0.0544 gives theta = arctan(0.0544 / 2) = arctan(0.0272) = 1.56 degrees. The included (cone) angle is twice the half-angle, so 3.12 degrees in this case.
What is the taper ratio format 1:X?
A taper ratio of 1:X means the diameter changes by 1 unit for every X units of axial distance. A ratio of 1:20 is therefore a very gentle taper (the Jarno taper standard uses exactly 1:20). You can calculate X from measured dimensions as X = L / (D1 - D2). Larger X means a shallower taper; smaller X means a steeper one.
What is the included angle and how is it different from the half-angle?
The half-angle is measured between the centreline of the workpiece and one sloped side of the taper. The included angle - also called the cone angle - is measured from one side of the taper to the other, passing through the apex. It is exactly twice the half-angle. When specifying angles for CNC programming or technical drawings, confirm whether the drawing calls for the half-angle or the included angle to avoid cutting a taper twice as steep or half as steep as intended.
Can I use this calculator for metric tapers?
Yes. Switch the unit selector to metric (mm). Enter D1, D2, and L in millimetres; all outputs update accordingly. The "taper per foot" output becomes "taper per metre" (TPm) because that is the analogous per-unit scale for metric: it tells you how many millimetres of diameter change occur per metre of axial length. Standard ISO metric tapers use ratio notation such as 1:20, 1:10, or 1:5; you can read the ratio directly from the taper ratio output.