Taper Calculator
A workpiece that steps down from a 2 inch diameter to a 1 inch diameter over an 8 inch length has a taper per foot of 1.5 inches, a taper ratio of 1 in 8, and a half angle of about 3.576 degrees. Use this taper calculator to convert a large diameter, small diameter, and the length between them into the four numbers machinists and woodworkers actually use: taper per foot, taper ratio, included angle, and half angle. Enter the two diameters and the length in the same unit, ideally inches, and the calculator returns all four values at once.
Quick answer
Taper per foot rescales the taper rate to inches of diameter change per 12 inches of length, and only makes sense when all three inputs are in inches.
Taper Per Foot
1.500in/ft
Taper Ratio
1 : 8
Half Angle
3.576°
Included Angle
7.153°
What this tells you
- •Taper per foot rescales the taper rate to inches of diameter change per 12 inches of length, and only makes sense when all three inputs are in inches.
- •Taper ratio is written as 1 : X, meaning the diameter changes by 1 unit over X units of length.
- •Half angle is the angle from the centerline to the tapered surface, the number you dial into a compound slide.
- •Included angle is twice the half angle, the full angle of the cone or wedge as seen from the side.
- •The large diameter must be greater than the small diameter, and the length must be positive.
How to Use
- 1Measure the large diameter (D) of the tapered section.
- 2Measure the small diameter (d) of the tapered section.
- 3Measure the length (L) between the two diameters, in the same unit as the diameters, ideally inches.
- 4Enter all three values and click Calculate.
- 5Read the taper per foot, taper ratio, half angle, and included angle from the result.
How It Works
Formula
Taper Per Foot = ((D - d) / L) x 12
Taper Ratio = 1 : (L / (D - d))
Half Angle = atan((D - d) / (2 x L)) x 180 / pi
Included Angle = 2 x Half AngleThe taper rate starts as the diameter difference (D minus d) divided by the length. Multiplying that rate by 12 converts it to inches of taper per foot of length, which only works when D, d, and L are all in inches. Dividing L by the same diameter difference gives the taper ratio as 1 unit of diameter change over that many units of length. The half angle comes from the arctangent of the diameter difference divided by twice the length, since half the diameter difference and the full length form the two legs of a right triangle along the taper. Doubling the half angle gives the included angle, the full angle of the cone as seen from the side.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
2-inch to 1-inch diameter over 8 inches
The diameter drops by 1 inch over 8 inches of length. That is a taper rate of 1/8, which becomes 1.5 inches per foot once multiplied by 12. The same 1/8 rate is a 1 : 8 taper ratio, a 3.576 degree half angle, and a 7.153 degree included angle.
1.5-inch to 1-inch diameter over 6 inches
The diameter drops by 0.5 inches over 6 inches of length, a taper rate of 1/12. That works out to exactly 1 inch of taper per foot, a 1 : 12 ratio, a 2.386 degree half angle, and a 4.772 degree included angle.
Morse taper style shank, 0.475-inch drop over 2.1 inches
A steep, short taper like this pushes the taper per foot and both angles much higher than a long, gentle taper does. The taper ratio still expresses the same geometry as 1 unit of diameter change over about 4.42 units of length.
Common Taper Ratio Reference
Approximate half angle and included angle for common taper ratios, assuming the ratio holds along the full length.
| Taper Ratio | Taper Per Foot | Half Angle | Included Angle |
|---|---|---|---|
| 1 : 4 | 3.000 in/ft | 7.125° | 14.250° |
| 1 : 8 | 1.500 in/ft | 3.576° | 7.153° |
| 1 : 12 | 1.000 in/ft | 2.386° | 4.772° |
| 1 : 16 | 0.750 in/ft | 1.790° | 3.580° |
| 1 : 20 | 0.600 in/ft | 1.432° | 2.864° |
| 1 : 50 | 0.240 in/ft | 0.573° | 1.146° |
Values are rounded. Self-holding Morse and Brown and Sharpe tapers usually fall near 1 : 20, while self-releasing tapers like Jarno and machine tool spindle tapers vary by standard.
How do taper per foot, taper ratio, and taper angle relate to each other?
All four numbers this calculator returns describe the same cone or wedge shape. They exist because different trades and machines prefer different formats. A drafter might specify a taper ratio on a drawing, a lathe operator might set a compound slide to a half angle, and a catalog might list taper per foot for a standard shank. Converting between them just moves the same diameter-to-length rate through different formulas.
Taper per foot answers the question, if this taper continued for a full 12 inches of length, how many inches would the diameter change? It is a rate, not a length, so it works for tapers shorter or longer than a foot. It only produces a meaningful number when the diameters and length are measured in inches, because the formula multiplies by 12 to rescale to a 12-inch basis. Enter millimeters or another unit and the taper-per-foot figure will not mean what its name implies.
Taper ratio is the most compact way to describe a taper because it strips out units entirely, as long as both diameters and the length use the same unit. A 1 : 8 taper means the diameter changes by 1 unit for every 8 units of length traveled along the axis. Smaller second numbers mean steeper tapers. A 1 : 4 taper is much steeper than a 1 : 50 taper.
Half angle is measured from the centerline of the part out to the tapered surface. It is the number stamped on many tooling references and the number a machinist dials directly into a compound slide protractor, because the slide only needs to swing to one side of center to cut the taper.
Included angle is the full angle you would measure with a protractor placed across the widest part of the taper, from one tapered surface to the other. It shows up on drawings that specify the total cone angle rather than the offset from center, and it is exactly double the half angle.
Standard machine tapers give a sense of scale. A Morse taper, common on drill and reamer shanks, runs close to a 1 : 20 ratio depending on the specific size. Jarno tapers use a simpler 1 : 20 rule across their whole range. Steeper self-releasing tapers used on some machine spindles can run near 3 : 4 or similar aggressive ratios so the tool releases cleanly instead of wedging in place.
Whether a taper self-holds or self-releases depends on the angle and the friction between the mating surfaces, not on this calculator. Shallow tapers, roughly under about 3 degrees of half angle, tend to wedge and hold under axial load. Steeper tapers tend to release more easily. This tool reports geometry only, not a holding-power prediction.
The length input should be the axial distance over which the diameter changes, measured along the centerline, not a diagonal or surface measurement. Using a surface-follow measurement instead of a straight axial length will skew every result, especially the angle figures.
Small measurement errors matter more on short, steep tapers than on long, shallow ones, because a small change in the diameter difference is a larger fraction of a short length. Measure diameters with a caliper or micrometer rather than a ruler whenever precision matters, and take the length reading along the same axis as the diameters.
This calculator assumes a straight, uniform taper along the full length, meaning the diameter changes at a constant rate from one end to the other. It does not model stepped, compound, or curved profiles, and it does not account for a workpiece that is not perfectly round or centered.
Common mistakes
- Mixing inches and millimeters between the diameters and the length
- Using taper per foot with non-inch inputs, where the per-foot scaling no longer means anything
- Measuring length along the tapered surface instead of along the centerline axis
- Entering radius values instead of full diameters
- Swapping the large and small diameter, which reverses the direction of the taper
- Confusing half angle with included angle when setting a compound slide or reading a drawing
- Assuming every taper ratio self-holds under load without checking the angle and application
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