Manual machining
Taper calculator
Taper per foot = 12 x (D - d) / L; set the compound to half the included angle.
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
Taper per foot
in/ft
TPF = 12 x (D - d) / L = 12 x (1.25 - 0.75) / 4 = 1.5000 in/ft
Taper per inch: 0.1250 in/in on diameter.
Half angle (compound setting)
°
half angle = atan((D - d) / (2 x L)) = atan((1.25 - 0.75) / (2 x 4)) = 3.5763°
Compound setting: 3° 34' 35". Included angle: 7.1527° (7° 9' 10").
Source: Machinery's Handbook (tapers; standard taper dimensions). Taper values computed from TPF.
How it works
TPF = 12 x (D - d) / L
| Symbol | Meaning | Unit |
|---|---|---|
| TPF | Taper per foot, change in diameter over 12 in of length | in/ft |
| D | Large end diameter | in |
| d | Small end diameter | in |
| L | Length of the taper along the axis | in |
Angles
half angle = atan((D - d) / (2 x L))
| Symbol | Meaning | Unit |
|---|---|---|
| half angle | Angle from the axis; the compound setting | degrees |
| D, d, L | As above, any length unit | in or mm |
Included angle = 2 x half angle. From a taper per foot: half angle = atan(TPF / 24). Taper per inch = (D - d) / L = TPF / 12. Small end: d = D - TPF x L / 12. In metric, a taper is written as a ratio 1:k with k = L / (D - d), and the half angle is atan(1 / (2 x k)).
Worked example: taper and compound setting
D = 1.250 in, d = 0.750 in, L = 4.000 in.
- TPF = 12 x (1.250 - 0.750) / 4.000 = 12 x 0.500 / 4.000 = 1.500 in/ft.
- Taper per inch: 0.500 / 4.000 = 0.1250 in/in.
- Half angle: atan(0.500 / 8.000) = atan(0.0625) = 3.5763°.
- Included angle: 2 x 3.5763 = 7.1527°.
Result: 1.500 in/ft; set the compound to 3.5763° (3° 34' 35").
Worked example: small end from taper per foot
D = 1.000 in, TPF = 0.600 in/ft, L = 3.000 in.
- Change over the length: 0.600 x 3.000 / 12 = 0.150 in.
- d = 1.000 - 0.150 = 0.850 in.
Result: small end 0.850 in (21.59 mm).
| Taper | Taper per foot (in/ft) | Included angle (°) | Included angle, D-M-S | Compound setting (°) |
|---|---|---|---|---|
| Morse #0 | 0.62460 | 2.9816 | 2° 58' 54" | 1.4908 |
| Morse #1 | 0.59858 | 2.8574 | 2° 51' 27" | 1.4287 |
| Morse #2 | 0.59941 | 2.8614 | 2° 51' 41" | 1.4307 |
| Morse #3 | 0.60235 | 2.8754 | 2° 52' 31" | 1.4377 |
| Morse #4 | 0.62326 | 2.9752 | 2° 58' 31" | 1.4876 |
| Morse #5 | 0.63151 | 3.0145 | 3° 0' 52" | 1.5073 |
| Morse #6 | 0.62565 | 2.9866 | 2° 59' 12" | 1.4933 |
| Morse #7 | 0.62400 | 2.9787 | 2° 58' 43" | 1.4894 |
| 7/24 steep taper (NMTB, CAT, BT) | 3.500 | 16.5943 | 16° 35' 39" | 8.2971 |
| Brown & Sharpe | nominal 1/2 in/ft (most sizes; per-size values differ) | 2.3870 | 2° 23' 13" | 1.1935 |
| R8 (TPF computed from the angle) | 3.5547 | 16.8500 | 16° 51' 0" | 8.4250 |
Included angle = 2 x atan(TPF / 24), computed from the taper per foot. R8 is listed by its included angle (16° 51') with the taper per foot computed from it. Brown & Sharpe: the angles use the nominal 1/2 in/ft (0.500 in/ft) that most sizes share; per-size values differ, so check the size you have.
Source: Machinery's Handbook (standard taper dimensions); angles computed
How to choose a setup
- Long, shallow tapers between centers: tailstock offset.
- Steeper or short tapers: a taper attachment or the compound.
- Short, steep tapers: set the compound to half the included angle.
Shop notes
- Indicate the finished taper or blue it against a gauge. The angle alone does not make the fit.
- A tailstock offset calculator for between-centers work is coming.
- TPF is measured on the diameter. The half angle is what the tool path makes with the axis.
FAQ
What angle do you set the compound to?
Half the included angle: 1.500 in/ft is 3.5763° (3° 34' 35").
What is the taper per foot of a Morse taper?
About 0.6 in/ft; MT2 is 0.59941 in/ft (2.8614° included).
What is a 7/24 taper in degrees?
16.5943° included, 3.500 in/ft.
How do you convert taper per foot to taper per inch?
Divide by 12: 1.500 in/ft = 0.1250 in/in.
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