Internal Taper Two-Ball Calculator: Angle, TPF & Mouth Diameter
In two-ball internal taper inspection, the sphere center depths from the datum face are z1 = h1 + r1 and z2 = h2 + r2, giving the exact half-angle via sin(α) = (r2 − r1) / (z1 − z2). With 12 mm and 20 mm balls at depths 38.5 mm and 12.2 mm, included angle is 20.6664° and mouth diameter is 28.425 mm.
Two-Ball Internal Taper Angle, TPF & Mouth Diameter Calculator
Compute internal cone half-angle from sin(α) = (r2 − r1) / (z1 − z2) where zi = hi + ri, included angle 2α, Taper Per Foot (TPF), and exact datum mouth diameter Dmouth = 2(r2/cos α + z2·tan α).
Why Does Two-Ball Taper Metrology Use sin(α) Instead of tan(α)?
Measuring an internal conical bore—such as a 7:24 CNC spindle socket (ASME B5.50 (V-Flange Tool Shanks)), a Morse taper sleeve, or a hydraulic valve seat—directly at the mouth with vernier calipers fails because a chamfered rim has no sharp, repeatable measuring edge. Dropping two Grade 5 tungsten-carbide gauge balls of diameters d1 and d2 into the bore and measuring depth h1 and h2 from the flat datum face to the top crest of each ball with a depth micrometer solves both the angle and the virtual mouth diameter with sub-micron repeatability. As proven in the conical metrology section of Machinery's Handbook (31st Ed., Industrial Press) and Pratt & Whitney metrology guides, a sphere inside a cone does not touch the wall at its horizontal equator; its circle of tangency lies perpendicular to the sloped cone wall.
Consequently, in the right triangle formed by the vertical distance between the two sphere centers Δz = z1 − z2 = (h1 + r1) − (h2 + r2) (the hypotenuse) and the difference in perpendicular ball radii r2 − r1 (the opposite leg), the exact trigonometric relation is sin(α) = (r2 − r1) / Δz—never tan(α)! Once half-angle α is found, projecting from the upper sphere center z2 to the datum plane z = 0 gives the exact mouth diameter Dmouth = 2 × (r2 / cos(α) + z2 × tan(α)), with length and angle conversions conforming to NIST SI Units Reference and NIST Special Publication 811.
| Internal Taper Bore | Ball Pair d1 / d2 | Micrometer Depths h1 / h2 | Included Angle 2α | TPF & Taper Ratio | Mouth Ø at Datum Face |
|---|---|---|---|---|---|
| Morse Taper MT2 Socket (1:20.020) | Ø10 / Ø16 mm | 50.100 / 10.070 mm (Δz=37.030) | 9.2939° (α=4.6469°) | 1.9508 in/ft (1:6.15) | Ø18.990 mm |
| Morse Taper MT3 Socket (1:19.922) | Ø14 / Ø22 mm | 64.500 / 12.695 mm (Δz=47.805) | 9.5995° (α=4.7997°) | 2.0152 in/ft (1:5.95) | Ø26.057 mm |
| Morse Taper MT4 Socket (1:19.254) | Ø18 / Ø28 mm | 82.000 / 13.735 mm (Δz=63.265) | 9.0659° (α=4.5330°) | 1.9027 in/ft (1:6.31) | Ø32.486 mm |
| 7/24 NMTB / CAT40 Spindle (3.500 TPF) | Ø20 / Ø35 mm | 56.400 / 11.989 mm (Δz=36.911) | 23.4473° (α=11.7236°) | 4.9805 in/ft (1:2.41) | Ø47.985 mm |
| 1:10 Precision Hydraulic Valve Bore | Ø12 / Ø20 mm | 48.000 / 12.025 mm (Δz=31.975) | 14.3728° (α=7.1864°) | 3.0261 in/ft (1:3.97) | Ø25.713 mm |
How Do You Select the Best Gauge Ball Pair to Minimize Angular Error?
Because any micrometer depth reading uncertainty δh is divided by the center span Δz in sin(α) = Δr / Δz, selecting two gauge balls that sit too close together amplifies small depth errors into large angle errors. Always choose d1 near the bottom of the taper and d2 near the mouth so that Δz ≥ 20 mm, and verify that Δz > r1 + r2 if both balls are placed in the cavity simultaneously.
| Inspection Condition | Depth h1 | Depth h2 | Computed 2α | Angular Error Δ(2α) | Computed Mouth Ø |
|---|---|---|---|---|---|
| Baseline measurement (Δh = 0.000 mm) | 38.500 mm | 12.200 mm | 20.6664° | — | 28.425 mm |
| +0.010 mm depth micrometer error on h1 | 38.510 mm | 12.200 mm | 20.6570° | -0.0094° | 28.421 mm |
| +0.025 mm (0.001") chip/burr under ball 1 | 38.525 mm | 12.200 mm | 20.6430° | -0.0234° | 28.415 mm |
| +0.005 mm ball diameter error on d2 | 38.500 mm | 12.200 mm | 20.6818° | +0.0154° | 28.438 mm |
Worked Example: Finding Included Angle, TPF, and Mouth Diameter
Given lower ball d1 = 12.000 mm (r1 = 6.000 mm) at depth h1 = 38.500 mm and upper ball d2 = 20.000 mm (r2 = 10.000 mm) at depth h2 = 12.200 mm, the sphere centers sit at z1 = 44.500 mm and z2 = 22.200 mm, giving center span Δz = 22.300 mm. From sin(α) = 4.000 / 22.300, the half-angle is 10.3332°, included angle is 20.6664°, Taper Per Foot is 4.3759 in/ft (ratio 1 : 2.74), datum mouth diameter is 28.425 mm, and small-end diameter at 50.0 mm depth is 10.192 mm. Compare external taper turning on our taper per foot calculator, check standard tapers on the Morse taper chart, measure dovetails on the dovetail over pins calculator, and build angle stacks with the sine bar calculator and gage block stack calculator.
Cite this page
ToolSpecVault Engineering Data Team. (2026, October 29). Internal Taper Two-Ball Calculator: Angle, TPF & Mouth Diameter. ToolSpecVault. https://toolspecvault.com/internal-taper-bore-two-ball-measurement-calculator/