Article — Taper Calculator (TPF, Angle, Ratio)
Taper calculator: TPF, taper ratio, and machine taper angle conversion
A taper with 1-inch large diameter, 0.75-inch small diameter, and 2-inch length has TPF = 1.5 in/ft, half-angle 3.576° (3° 35'), and ratio 1:8. Morse taper MT2 runs TPF ≈ 0.6 in/ft (1:20 ratio). NPT pipe thread is exactly 1:16 (TPF = 0.75). Half-angle equals arctan((D-d)/(2L)).
Tapers are everywhere in machining. Drill chucks, lathe tailstocks, CNC tool holders, pipe fittings, and reamers all use precision tapers to hold tools or seal connections. The math is direct — three numbers in, six numbers out — but the variety of conventions (TPF, TPI, ratio, half-angle, included angle) makes the calculation tedious to do by hand for every job. This calculator does it once.
What is a machine taper?
A machine taper is a precision cone that fits into a matching socket to hold a tool or workpiece. Tapers are described by three dimensions: the diameter at each end (large D and small d) and the length over which the diameter changes (L). From these, every other taper measurement derives.
Self-holding tapers (Morse, Brown & Sharpe, Jacobs) have shallow angles around 1-2°. The shallow angle creates enough friction to hold the tool without external clamping. Quick-release tapers (CAT, BT, HSK) have steeper angles around 8° (7/24 ratio for CAT) and require a drawbar or hydraulic clamp because they cannot self-lock.
The taper per foot formula
Taper per foot (TPF) is the diameter change per foot of taper length. The formula: TPF = (D - d) × 12 / L, with all dimensions in inches. For D = 1.0 in, d = 0.75 in, L = 2 in: TPF = 0.25 × 12 / 2 = 1.5 inch per foot. Most precision machine tapers run TPF 0.5 to 0.75 in/ft; pipe threads run higher; CNC tool tapers run much higher.
Taper per inch (TPI) is the same physical quantity in different units: TPI = (D - d) / L. The 12× factor converts between them. Machinists use TPF because the numbers are larger and easier to compare; engineers and CAD software often use TPI or the 1:N ratio. The calculator outputs all three forms.
Stephen A. Morse developed his taper system in 1864 while working at Morse Twist Drill in New Bedford, Massachusetts. He chose roughly 5/8 inch per foot — not for any mathematical reason, but because it created enough friction to hold a drill securely while still being easy to remove with a drift key. The same approximate ratio remains the international standard 160 years later.
Machine taper half-angle
The half-angle (α) is the angle between the taper centerline and the cone surface. It is computed as arctan((D - d) / (2L)). For our example with D = 1.0, d = 0.75, L = 2: α = arctan(0.0625) = 3.576° = 3° 35'. The half-angle is the practical setting for any taper turning operation.
The included angle (θ) is twice the half-angle. It is the full cone angle visible from the side. Some specifications quote the included angle, others quote the half-angle — always confirm which when reading a print. Reamer specifications usually use included angle; lathe taper attachments usually use half-angle.
For very small tapers, the small-angle approximation gives half-angle ≈ (D - d) / (2L) in radians, converted to degrees by multiplying by 57.296. For Morse MT2 with TPF ≈ 0.6: half-angle ≈ 0.025 / 2 in radians × 57.296 = 1.43°. The approximation is good to 0.01° for any taper under 5°.
Morse taper standards
Morse tapers come in nine sizes: MT0 through MT7 plus the half-size MT4.5. The nominal TPF varies slightly between sizes — MT1 and MT2 use 5/8 in/ft (0.625), MT3 uses 0.6024, MT4 uses 0.6233, and so on. The variation is historical, arising from different commercial conventions when the standard was unified into ISO 296.
MT2 is the most common Morse taper in benchtop machinery. Drill press tailstocks and small lathe spindles use MT2 almost universally. Larger industrial lathes use MT4 or MT5. Each size has a specific large diameter and length, so MT2 and MT3 tools are not interchangeable even though both are Morse tapers.
Both are self-holding tapers around 0.5-0.6 TPF, but Morse and Brown & Sharpe use different specific dimensions. A Brown & Sharpe #2 will not fit a Morse #2 socket and vice versa. The tapers were developed separately by competing manufacturers in the 1860s-1870s. Always identify the specific standard before swapping a tool — there are gauges to test which type your machine uses.
CNC taper systems
Modern CNC machines use steep-angle tool holders that require a drawbar to retain the tool. The most common is CAT (Caterpillar) or BT (Mori Seiki) taper with 7/24 in/ft = 3.5 TPF, which gives a half-angle of about 8.3°. CAT/BT-40 fits most mid-size CNC mills; CAT/BT-50 fits larger machines.
HSK (Hohlschaftkegel) is the European high-speed alternative, with 1:10 taper (TPF = 1.2 in/ft, half-angle 2.86°). The HSK design adds a face contact between the tool and spindle, dramatically improving rigidity and repeatability at high spindle speeds (above 10,000 RPM). HSK-63 is common on 5-axis machining centers.
Taper turning tailstock offset
To turn a taper between centers on a lathe, shift the tailstock perpendicular to the bed by an amount equal to stock length times sine of the half-angle: offset = L_stock × sin(α). For a 12-inch stock at 2° half-angle: offset = 12 × 0.0349 = 0.419 inch. Verify with a test cut and dial indicator before machining the final piece.
The tailstock-offset method has a limit: stock can only be tapered about 8-10° this way before the centers no longer engage the centerline of the part. Steeper tapers require a taper attachment (a special bracket that swings the cross-slide) or a compound rest set at the half-angle. The compound rest method is the standard for short, steep tapers like NPT pipe threads.
When checking a taper for fit, use bluing dye (Prussian blue) on the inner socket and insert the taper. Pull it out and inspect the surface — even bluing transfer means good contact. Spots of no transfer indicate high points that need lapping. This technique catches taper errors that exceed the visible-light resolution of digital calipers.
Common taper calculation mistakes
Mistake one is confusing half-angle with included angle. A print specifying "7° taper" might mean either 7° half-angle (14° included) or 7° included (3.5° half-angle). The calculator outputs both — always check which the print uses before setting up the machine.
Mistake two is computing TPF from one diameter and length without measuring the small end. The formula needs both D and d. If you only have one diameter and the angle, use the trigonometric relation: small d = D - 2L × tan(half-angle). For D = 1 in, L = 2 in, half-angle = 2°: d = 1 - 2 × 2 × 0.0349 = 0.860 in.
Mistake three is using metric dimensions but quoting TPF in inches per foot. TPF is dimensionless in the sense that the units cancel — but the formula assumes consistent length units throughout. If D, d, and L are in mm, the result is mm per foot of length only if you multiply by 12 anyway. Better to convert to inches first or to use the 1:N ratio form, which is truly unitless.
TPI (D - d) / LTPF TPI × 12Half-angle arctan((D-d)/2L)1:N L / (D - d)NPT pipe 1:16 = 0.75 TPFMorse typical 1:20 ≈ 0.6 TPF- TPF = (D-d) × 12 / L standard taper per foot formula
- Half-angle = arctan((D-d) / 2L) setting for sine bar and compound rest
- 1:20 approximate Morse taper ratio
- 1:16 exact NPT pipe thread ratio
- 7/24 in/ft (3.5 TPF) standard CAT/BT CNC taper
- 1:10 HSK high-speed CNC taper
- 1864 year Stephen Morse standardized the Morse taper
- ±1-2 minutes ISO 296 tolerance for Morse tapers