Calculate taper angle, taper ratio and taper per foot from the large diameter, small diameter and length, or solve a missing diameter or taper length.
Taper Calculator Formulas and Usage
The taper calculator has five explicit modes: calculate taper from two diameters and axial length; solve the major diameter, minor diameter, or axial length using a half-angle; or convert a diameter taper-rate convention. These relationships describe a straight circular taper whose diameter changes uniformly over its axial length.
In practical terms, taper describes how quickly a diameter increases or decreases over distance. A larger taper means a faster change in size over a shorter length, while a smaller taper means a gentler transition.
Core Taper Equations
For a straight taper, the diameter change is spread evenly across the total axial length.
Here, α is the taper half-angle measured from the centerline to one side of the taper, and θ is the included angle measured across the full cone. If you are comparing the result to a drawing, make sure the drawing specifies whether it uses half-angle or included angle.
Variable Definitions
| Symbol | Meaning | Notes |
|---|---|---|
| D | Larger diameter | The diameter at the wide end of the taper |
| d | Smaller diameter | The diameter at the narrow end of the taper |
| L | Taper length | Measured along the axis, not along the sloped surface |
| T | Taper rate | Change in diameter per unit of length |
| α | Taper angle | Half-angle from the centerline to the surface |
| θ | Included angle | Full angle from one side of the taper to the other |
Rearranged Formulas
The rearranged equations below can use a known diameter taper rate T. The calculator’s diameter and length solve modes instead take the half-angle α, with T = 2 tan α. If you know a rate, use Convert taper-rate convention to find the corresponding half-angle first.
| Find | Formula |
|---|---|
| Larger diameter | D = d + TL |
| Smaller diameter | d = D - TL |
| Length | L = (D - d) / (T) |
| Taper ratio | n = (L) / (D - d) = (1) / (T) |
The taper ratio is often written as 1:n. A taper of 1:20 means the diameter changes by 1 unit for every 20 units of axial length.
How to Use the Calculator
- Choose the quantity or conversion you want to solve.
- Enter the visible required diameters and axial length, using mm or in beside each field. Mixed units are converted to the same basis.
- For a diameter or length inverse, enter the half-angle from the centerline in degrees, from 0° up to but not including 90°.
- For rate conversion, select diameter change per unit length, mm or in per foot or metre, or 1:N. Enter N for the ratio convention.
- Select mm or in for linear results where offered. Angles are reported in degrees.
- Select Calculate and review the answer and calculation details. Reset clears this calculator’s fields and saved browser snapshot.
For accurate results, the large and small diameters must refer to the same feature type. For example, use outside diameters together or inside diameters together, and measure the taper length along the centerline of the part.
Important Interpretation Notes
- Taper is not the same as angle. Taper is a rate of diameter change, while angle describes the slope of the side wall.
- The angle shown by these formulas is the half-angle. If your print calls for the full included angle, double the result.
- Consistent units matter. The calculator converts each selected mm or in input before applying the equations. T is diameter change divided by axial length in the same unit; taper per foot or metre also states the diameter unit explicitly.
- A zero taper means no change in diameter. Equal positive diameters give a cylinder with a zero half-angle. Its 1:N ratio has no finite denominator. Equal diameters with a zero half-angle leave the positive axial length undetermined.
- Check diameter order. Major diameter must be positive and at least the nonnegative minor diameter. A zero minor diameter is a cone tip; axial length must be positive.
Examples
If the larger diameter is 10 in, the smaller diameter is 5 in, and the taper length is 3 in, the taper rate is:
The taper half-angle is:
The included angle is:
If a part must go from 2.50 in to 2.00 in with a taper of 0.125 per inch, the required taper length is:
So the taper must extend over 4 inches of axial length. In the calculator, convert T = 0.125 per unit length to its half-angle (about 3.576334°), then use Axial length with the two diameters and that half-angle. Keep as much angle precision as available when a tolerance matters; the displayed values are rounded.
Where Taper Calculations Are Commonly Used
- Lathe turning and CNC machining
- Tool holders and machine tapers
- Pipe reducers and nozzles
- Shaft transitions and mechanical fits
- Molded parts and conical components
- Structural or fabrication transitions between two diameters
Frequently Asked Questions
Is taper measured on the diameter or the radius?
In these formulas, taper is based on the change in diameter. That is why the angle equation uses 2L in the denominator. Each side of the taper changes by half of the total diameter difference.
What is the difference between taper and taper angle?
Taper is the amount of diameter change per unit length. Taper angle is the geometric slope of the side wall. Two tapers with different units can describe the same physical shape, but the angle stays tied to the actual geometry.
Why does the calculator need the larger and smaller diameters?
The total diameter reduction is the starting point for all taper calculations. Once that change is known, the taper rate and angle can be determined from the length.
Can I use the calculator for inside tapers?
Yes. The same equations apply as long as both diameters are internal diameters and the length is measured along the axis of the tapered section.
What if I only know the angle and one diameter?
You still need enough information to define the full geometry. In most cases that means knowing one more independent value, such as the second diameter or the taper length.
