Taper calculator
Setting angle, included angle and taper ratio from three known dimensions. With Morse and steep tapers as presets and the angle in degrees, minutes and seconds.
The setting angle is half the included angle. It is the angle the compound slide is swivelled by. The sketch is schematic, only the ratio of the diameters is to scale. A standard taper sets the taper ratio and, where standardised, the large diameter.
Formulas at a glance
How to calculate the setting angle
D large diameter, d small diameter, L taper length. Example: D 40, d 30, L 100 gives tan(α/2) = 0.05, so a setting angle of 2.8624° or 2° 51′ 45″.
How to calculate the taper ratio
The taper ratio states over what length the diameter changes by 1 mm. For D 40, d 30, L 100 it is 1 : 10.
Calculating the small diameter
When angle, large diameter and length are known. The large diameter uses the same formula with plus instead of minus.
Calculating the taper length
When both diameters and the angle are known. With a 1 : 10 taper and 10 mm difference in diameter it is 100 mm.
Standard tapers at a glance
Morse tapers to DIN 228, 7 : 24 steep tapers to ISO 7388, plus metric taper, HSK and taper pin. The angles are calculated from the taper ratio. D is the large diameter at the gauge plane. MK is the German designation for Morse taper (MT), SK for steep taper.
| Taper | Ratio | Included angle | Setting angle | D (mm) |
|---|---|---|---|---|
| MK0Morse taper | 1 : 19.212 | 2.9816° | 1.4908°1° 29′ 27″ | 9.045 |
| MK1Morse taper | 1 : 20.047 | 2.8575° | 1.4287°1° 25′ 43″ | 12.065 |
| MK2Morse taper | 1 : 20.02 | 2.8613° | 1.4307°1° 25′ 50″ | 17.780 |
| MK3Morse taper | 1 : 19.922 | 2.8754° | 1.4377°1° 26′ 16″ | 23.825 |
| MK4Morse taper | 1 : 19.254 | 2.9751° | 1.4876°1° 29′ 15″ | 31.267 |
| MK5Morse taper | 1 : 19.002 | 3.0146° | 1.5073°1° 30′ 26″ | 44.399 |
| MK6Morse taper | 1 : 19.18 | 2.9866° | 1.4933°1° 29′ 36″ | 63.348 |
| SK30Steep taper | 1 : 3.429 | 16.5943° | 8.2971°8° 17′ 50″ | 31.750 |
| SK40Steep taper | 1 : 3.429 | 16.5943° | 8.2971°8° 17′ 50″ | 44.450 |
| SK50Steep taper | 1 : 3.429 | 16.5943° | 8.2971°8° 17′ 50″ | 69.850 |
| ME 1:20Other | 1 : 20 | 2.8642° | 1.4321°1° 25′ 56″ | – |
| HSK 1:10Other | 1 : 10 | 5.7248° | 2.8624°2° 51′ 45″ | – |
| 1:50Other | 1 : 50 | 1.1459° | 0.5729°0° 34′ 23″ | – |
Included angle, setting angle, taper ratio
A taper can be specified in three ways. The included angle α is the full angle between the two sides. The setting angle α/2 is half of it, the angle between the side and the axis. The taper ratio 1 : x states over how many millimetres of length the diameter grows by one millimetre.
Drawings usually give the taper ratio or the included angle. On a manual lathe you need the setting angle, because the compound slide is swivelled by that amount. The most common mistake is setting the full included angle. The taper then comes out twice as steep.
From the shop floor: Tool tapers such as the Morse taper are not set by the scale, which is too coarse. Align the compound slide with a dial gauge against a test bar or an existing socket and check the result with marking blue on a taper gauge. The contact pattern should be even over the whole length.
Frequently asked questions
How do you calculate the setting angle for a taper?
The tangent of the setting angle is large diameter minus small diameter, divided by twice the taper length. For D 40, d 30 and L 100 that is 10 divided by 200, i.e. 0.05. The setting angle is therefore 2.8624° or 2° 51′ 45″.
What does a 1 : 10 taper mean?
Over 10 mm of length the diameter changes by 1 mm. The included angle is 5.7248°, the setting angle 2.8624°. The HSK taper has this ratio.
What is the difference between included angle and setting angle?
The included angle is the full angle between the sides, the setting angle is half of it. The compound slide is always set to the setting angle.
What angle does a Morse taper have?
It depends on the size, because each Morse taper has its own ratio. The setting angle lies between 1.4287° for MT1 and 1.5073° for MT5. MT3 has 1 : 19.922 and therefore 1.4377° or 1° 26′ 16″.
What angle does a steep taper have?
All steep tapers have the ratio 7 : 24. The included angle is 16.5943°, the setting angle 8.2971° or 8° 17′ 50″. Sizes 30, 40 and 50 differ only in diameter: 31.75 mm, 44.45 mm and 69.85 mm.
How do you check a turned taper?
With a taper gauge and marking blue. Apply the blue thinly to the taper, fit the gauge and turn it slightly. If the blue transfers evenly over the whole length, the angle is right. If it only marks at the front or the back, the setting angle needs correcting.
Software for manufacturers
We built these calculators ourselves. For companies we develop Agentority, an AI that runs on your own server and works with your ERP, inbox and warehouse. It drafts quotes from customer emails, flags delivery delays and books in delivery notes from a photo.
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