Calculate taper per unit length, taper ratio, and half-angle from two diameters and axial length.
Enter the major diameter, minor diameter, and axial length, then calculate the taper.
The major diameter must be larger than the minor diameter after unit conversion.
This taper calculator describes how a round part narrows from a major diameter to a minor diameter over an axial length. It returns taper per unit length, the equivalent “1 in N” ratio, and the half-angle between the surface and the centerline.
It suits straight conical sections such as shafts, pins, plugs, and turned work. It does not account for curved profiles, tolerances, surface finish, tool deflection, or a drawing’s particular taper standard.
The major diameter must exceed the minor diameter and the length must be positive. Because all dimensions are normalized, the rate itself has no length unit.
T = (D − d) ÷ L
N = L ÷ (D − d), so the taper ratio is 1 in N
Half-angle θ = arctan[(D − d) ÷ (2L)]
The included cone angle is 2θ. The factor of two appears because the radial change is half the diameter change.
| Symbol or result | Meaning |
|---|---|
| D | Major, or larger, diameter. |
| d | Minor, or smaller, diameter. |
| L | Axial distance over which diameter changes. |
| T | Diameter change per unit axial length. |
| θ | Angle from centerline to one tapered surface. |
| 2θ | Included angle between opposite surfaces. |
For D = 30 mm, d = 20 mm, and L = 100 mm, the diameter change is 10 mm. T = 10/100 = 0.1 per unit, and N = 100/10 = 10, so the taper is 1 in 10.
The half-angle is arctan(10/(2 × 100)) = 2.862405°. The included angle is about 5.72481°.
T = 0.1 means the diameter changes by 0.1 unit for every one unit of axial travel. Because numerator and denominator use the same dimension after conversion, 0.1 mm/mm is the same geometric rate as 0.1 in/in.
“1 in 10” here refers to one unit of diameter change over ten units of axial length. Some drawings use taper on diameter, while others discuss slope on one side or specify an included angle. Confirm the convention before machining or inspection.
The displayed angle is the half-angle to the axis. A setup requiring the full included cone angle needs twice this value. Confusing those two is a common source of parts cut at the wrong angle.
Mixed units are acceptable because each length is converted first. Still, entering all measurements in the drawing’s unit makes transcription easier to audit. Never attach a length unit to the dimensionless rate itself.
Real dimensions carry tolerances. Maximum and minimum diameters and length can produce a range of acceptable taper values. For inspection, use the method and datum specified on the engineering drawing.
A nominal taper does not determine tool offsets, feed, material allowance, or fit by itself. Machine geometry, compound-rest convention, CNC coordinates, gauge method, and thermal conditions all affect shop practice.
As a check, multiply T by L; it should reproduce D − d. Also, multiplying the radial slope tan θ by two should recover the taper per length.
For a shallow taper, the half-angle in radians is close to half the taper rate, but this small-angle approximation becomes less accurate as the angle grows. The calculator uses arctangent rather than that approximation.
The diameter changes by one unit over ten units of axial length. Confirm that your drawing also defines taper on diameter rather than one-side slope.
No. It is the half-angle between one surface and the centerline. Double it to obtain the included angle between opposite surfaces.
Yes. The page converts each to metres before comparing them. Correct unit selection remains essential for a meaningful result.
The form describes narrowing from D to d. Reversing them would produce a negative signed taper and would contradict the labeled inputs.
It is a ratio of two lengths and is dimensionless after unit conversion. You may describe it as millimetres per millimetre or inches per inch for clarity.
Use axial length parallel to the centerline between the two diameter locations, not the longer distance along the sloped surface.
Use the part drawing, tolerances, machine convention, and an appropriate inspection method. The geometric result alone does not establish a complete machining setup.