Taper Calculator
Work out a taper from its measurements. Enter the large and small diameters, the taper length and the workpiece length — get the taper per foot and per inch, the taper ratio, the included and per-side angles, and the lathe tailstock set-over, with a comparison to standard machine tapers (Morse, B&S, Jarno) and a downloadable PDF.
| Taper | Taper / foot | Incl. angle | vs yours |
|---|---|---|---|
| Enter a taper to compare | |||
| Length | Taper / foot | Angle / side | Incl. angle |
|---|---|---|---|
| Enter a taper to compare | |||
How to use the taper calculator
Four quick steps turn a taper’s dimensions into its full geometry — the taper per foot and per inch, the ratio, the included and per-side angles, and the lathe tailstock set-over, with a standard-taper comparison and a downloadable report.
Enter the diameters
The large and small end diameters of the taper, in inches.
Enter the lengths
The taper length and the total workpiece length (for the set-over).
Compare a standard
Optionally check against Morse, Brown & Sharpe, Jarno or NMTB tapers.
Read results & save a PDF
Get the taper per foot, angles, ratio and set-over — then download a report.
The formulas behind the taper
The diameter difference over the length gives the taper rate; the angle, ratio and tailstock set-over all follow from the same dimensions.
▶ Worked example
A taper from 1.0 in to 0.75 in over a 3 in length, on a 6 in part.
The difference is 0.25 in, so TPI = 0.25 ÷ 3 = 0.083 and TPF = 0.083 × 12 = 1.0 in/ft.
TPF 1.0 in/ft · ratio 1:12 · included angle 4.77° (per side 2.39°)
The tailstock set-over for turning between centers is 0.25 × 6 ÷ 6 = 0.25 in.
Taper quick reference
The core formulas, the standard machine tapers and their angles, and the taper per foot and angle for common ratios.
| Quantity | Formula |
|---|---|
| Taper per inch | (D − d) ÷ taper length |
| Taper per foot | taper per inch × 12 |
| Angle per side | atan((D − d) ÷ (2 × length)) |
| Included angle | 2 × angle per side |
| Taper ratio | length ÷ (D − d) |
| Tailstock set-over | (D − d) × total length ÷ (2 × taper length) |
| Taper | Taper per foot | Included angle |
|---|---|---|
| Morse MT0 | 0.6246 in/ft | 2.98° |
| Morse MT1 | 0.5986 in/ft | 2.86° |
| Morse MT2 | 0.5994 in/ft | 2.86° |
| Morse MT3 | 0.6023 in/ft | 2.88° |
| Morse MT4 | 0.6233 in/ft | 2.98° |
| Morse MT5 | 0.6315 in/ft | 3.01° |
| Brown & Sharpe | 0.5020 in/ft | 2.40° |
| Jarno | 0.6000 in/ft | 2.86° |
| NMTB / CAT | 3.5000 in/ft | 16.59° |
Self-holding tapers (Morse, B&S, Jarno) grip by friction; the steep NMTB/CAT releases and needs a drawbar.
| Ratio | Taper per foot | Included angle |
|---|---|---|
| 1 : 8 | 1.500 in/ft | 7.15° |
| 1 : 10 | 1.200 in/ft | 5.72° |
| 1 : 12 | 1.000 in/ft | 4.77° |
| 1 : 16 | 0.750 in/ft | 3.58° |
| 1 : 20 | 0.600 in/ft | 2.86° |
| 1 : 24 | 0.500 in/ft | 2.39° |
A larger second number is a shallower taper. Enter your own diameters and length above for any taper.
Everything the calculator gives you
One taper produces a full geometry profile, not just a single number.
Taper reference numbers
Built for anyone machining a taper
Machinists, engineers, and repair techs all need a taper’s rate, angle and set-over.
Cutting a taper on a lathe and needing the compound-rest angle or the tailstock set-over from the part’s dimensions.
- Set the compound to the per-side angle
- Offset the tailstock by the set-over
- Read the taper per foot and ratio
Specifying a taper on a drawing, or matching a feature to a standard machine taper so it mates with existing tooling.
- Convert diameters and length to TPF and angle
- Check a taper against Morse, B&S or Jarno
- Hand a reviewer a clean PDF of the figures
Identifying an unknown taper on a spindle, arbor or shank by measuring it and comparing against the common standards.
- Measure the diameters and length
- Compare the taper per foot to the standards
- Confirm the match before ordering tooling
Get the taper right
Small habits make a taper cut far more accurate.
Work from diameters
Taper per foot is the diameter difference over the length, times 12 — use diameters, not radii.
Compound to per-side angle
Set the compound rest to the per-side angle, not the included angle, for a short taper.
Set-over uses total length
The tailstock set-over depends on the whole part length, not just the tapered section.
Match a standard first
If it needs to mate with tooling, check it against Morse, B&S or Jarno before cutting.
Shallow grips, steep releases
Self-holding tapers hold by friction; steep NMTB/CAT tapers need a drawbar.
Check with a gauge
Verify the fit with a taper gauge and bluing before calling it done.
Confirm the drawing
Standard-taper values are nominal — cut to the part drawing and the actual gauge.
Frequently asked questions
Common questions about tapers, taper per foot, angles, and standard machine tapers.
A taper is worked out from the two end diameters and the length over which they change. The taper per inch is the large diameter minus the small diameter, divided by the taper length. Multiply by 12 to get the taper per foot.
From the same figures you get the angle, the ratio and, for turning between centers, the tailstock set-over. The large diameter must be bigger than the small one over a positive length.
For example, a taper from 1.0 to 0.75 inch over 3 inches has a taper per inch of 0.25 divided by 3, which is 0.083, or 1.0 inch per foot. This calculator returns the taper per foot and inch, the angles, the ratio and the set-over.
Taper per foot (TPF) and taper per inch (TPI) both describe how fast the diameter changes along the taper. TPI is the change in diameter per inch of length; TPF is the change per foot, which is just the TPI times 12.
So a taper that loses 0.083 inch of diameter per inch of length has a taper per inch of 0.083 and a taper per foot of 1.0. Machine tapers are usually quoted in taper per foot — Morse tapers are all close to 0.6 inch per foot.
Both are a measure of the whole diameter change, not the radius. This calculator reports the taper in both per-inch and per-foot form.
The taper angle comes from the diameters and the length. The angle per side — the angle of the surface from the axis — is the inverse tangent of the diameter difference divided by twice the length. The included angle, between the two sides, is twice that.
The division by two is because the radius, not the diameter, changes along each side. So a shallow taper has a small angle and a steep taper a large one.
For example, a 1.0 to 0.75 inch taper over 3 inches has a per-side angle of about 2.4 degrees and an included angle of about 4.8 degrees. This calculator gives both, in decimal degrees and degrees-and-minutes.
When you turn a taper between centers on a lathe by offsetting the tailstock, the set-over is the diameter difference times the total length of the part, divided by twice the taper length. It shifts the tailstock center sideways so the part turns at an angle to the tool.
The total length matters because the offset is applied over the whole part, not just the tapered section, so a longer part needs a bigger set-over for the same taper.
For example, a 1.0 to 0.75 inch taper over 3 inches on a 6-inch part needs a set-over of 0.25 times 6 divided by 6, which is 0.25 inch. This calculator works out the set-over from your dimensions.
The taper ratio expresses the taper as 1 unit of diameter change over so many units of length — written as 1 to a number. It is the taper length divided by the diameter difference.
So a taper whose diameter changes by 1 inch over 12 inches of length is a 1:12 taper. A larger second number means a shallower, more gradual taper.
For example, a 1.0 to 0.75 inch taper over 3 inches has a ratio of 3 divided by 0.25, which is 1:12. This calculator reports the taper ratio alongside the taper per foot and the angle.
These are standard self-holding machine tapers, used to hold tools and arbors in spindles and tailstocks. Morse (MT) tapers, in sizes MT0 through MT7, are the most common and run close to 0.6 inch per foot. Brown & Sharpe tapers are about 0.5 inch per foot, and Jarno tapers are exactly 0.6 inch per foot.
They are shallow tapers that grip by friction, so a tool seats and holds without a drawbar. Each size has defined diameters and lengths from its standard.
This calculator lets you compare your measured taper against these standards to see how close it is, which helps identify an unknown taper or check a cut one.
Self-holding tapers, like Morse and Brown & Sharpe, are shallow — around half an inch per foot — so they wedge in and grip by friction, holding a tool without a drawbar. They are used for drill arbors, tailstock tooling and the like.
Self-releasing tapers, like the NMTB and CAT tapers at 3.5 inches per foot, are much steeper. They locate precisely but do not grip, so they need a drawbar or retention knob to hold the tool in the spindle — common on milling machines.
So the taper angle tells you how it is held. This calculator reports the taper per foot, which is how these standards are distinguished, and compares your taper against them.
There are a few ways, each matched to the taper. For a short, steep taper, swivel the compound rest to the per-side angle and feed by hand. For a long, shallow taper between centers, offset the tailstock by the set-over. A taper attachment guides the cross-slide for a precise taper without offsetting centers.
Whichever method, you set the angle or offset from the taper’s dimensions — the per-side angle for the compound, the set-over for the tailstock.
This calculator gives the per-side angle for the compound rest and the tailstock set-over for turning between centers, so you can dial in whichever method your lathe uses. Always check the result against a taper gauge.
Disclaimer: US units (inches, degrees). Taper per inch = (large − small) ÷ taper length; taper per foot = taper per inch × 12. Angle per side = atan((D − d) ÷ (2 × length)); included angle is twice that. Tailstock set-over = (D − d) × total length ÷ (2 × taper length), for turning a taper between centers. Standard-taper values (Morse, Brown & Sharpe, Jarno) are nominal references and vary slightly by size and standard. A planning aid only — confirm against the part drawing and a taper gauge before machining.