What this calculator does
A taper calculator works out how quickly a round workpiece narrows along its length, given the diameter at each end and the distance between those two measurements. This is the standard machining and woodturning sense of taper: a shaft, pin, table leg or fitted joint that changes diameter steadily over its length.
Taper is usually stated one of two ways: as a ratio such as 1:10, meaning the diameter changes by 1 unit for every 10 units of length, or as an included angle in degrees, the total angle between the two tapering sides. This calculator gives both from the same three measurements, since different trades and drawings favour one convention or the other.
The formula
Taper is the difference between the two diameters divided by the length between them. The same figure restated as a ratio is the length divided by that diameter difference, written as 1 to that number. The included angle is twice the arctangent of the diameter difference over twice the length, since the taper runs symmetrically to the centre axis on each side.
| Term | Meaning |
|---|---|
| Taper | The diameter difference divided by length: (D − d) ÷ L. |
| Included angle | The full angle between the two tapering sides, as measured across the workpiece. |
| Half angle | The angle of one tapering side to the centre axis, half the included angle. |
The inputs explained
| Field | What to enter |
|---|---|
| Large diameter (mm) | The larger diameter of the tapered section, in millimetres. |
| Small diameter (mm) | The smaller diameter of the tapered section, in millimetres. |
| Length between diameters (mm) | The length of the tapered section between where those two diameters occur, in millimetres. |
When to use it
Checking a taper against a standard callout
Machine tapers are often specified as a standard ratio, such as 1:10 or Morse taper values; measuring the actual diameters and length on a part and computing its taper checks whether it matches the intended specification.
Setting up a lathe compound slide
Turning a taper on a lathe usually means setting the compound rest to the half angle, which this calculator gives directly from the two diameters and the length rather than needing a separate trigonometry step.
Designing a tapered joint or leg
Furniture and joinery tapers, such as a table leg narrowing from top to bottom, can be checked for a consistent, deliberate taper ratio rather than an accidental or uneven one.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How taper ratio and angle change with diameter difference at a fixed length
A fixed 200 mm length and 50 mm large diameter, across a range of small diameters.
| Small diameter | Taper | Taper ratio | Included angle |
|---|---|---|---|
| 45 mm | 0.0250 | 1 : 40.00 | 1.43° |
| 40 mm | 0.0500 | 1 : 20.00 | 2.86° |
| 35 mm | 0.0750 | 1 : 13.33 | 4.30° |
| 30 mm | 0.1000 | 1 : 10.00 | 5.72° |
| 25 mm | 0.1250 | 1 : 8.00 | 7.15° |
| 20 mm | 0.1500 | 1 : 6.67 | 8.58° |
How taper ratio and angle change with length at a fixed diameter difference
A fixed 20 mm diameter difference, spread over a range of lengths.
| Length | Taper | Taper ratio | Included angle |
|---|---|---|---|
| 100 mm | 0.2000 | 1 : 5.00 | 11.42° |
| 150 mm | 0.1333 | 1 : 7.50 | 7.63° |
| 200 mm | 0.1000 | 1 : 10.00 | 5.72° |
| 300 mm | 0.0667 | 1 : 15.00 | 3.82° |
| 400 mm | 0.0500 | 1 : 20.00 | 2.86° |
| 500 mm | 0.0400 | 1 : 25.00 | 2.29° |
Questions
What does a taper of 1:10 actually mean?
It means the diameter changes by 1 unit for every 10 units of length along the taper. A 1:10 taper over 100 mm of length would change diameter by 10 mm across that length.
Should I use the included angle or the half angle when setting a lathe?
A lathe compound slide is set to the half angle, since the tool moves along one side of the taper relative to the axis. The included angle is the full angle you would measure across the finished workpiece with a protractor.
Does this work for a taper measured in inches?
Yes, the taper ratio and angle come out the same regardless of unit, as long as the two diameters and the length are all entered in the same unit.
Is this the same as a running taper before a race?
No. This calculator covers the machining and engineering sense of taper, a workpiece changing diameter over its length. Reducing training load before a race is a different, more subjective use of the same word and has no fixed formula to calculate.
For a cylinder with a constant diameter rather than a tapering one, see the cylinder calculator. For the angle between two known sides of a triangle instead of a tapering diameter, see the trigonometric functions calculator.