BIZOMAT MAKEOS · Quality
Internal Taper Angle from Two Balls
What is the included angle and top diameter of this tapered bore, measured with two balls?
Included angle and diameter at the top face of an internal taper from the depths to the tops of two precision balls of different sizes.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Small ball diameter | mm | Measured diameter of the smaller precision ball (it sits deeper). |
| Large ball diameter | mm | Measured diameter of the larger ball. |
| Depth from top face to top of small ball | mm | Depth gauge reading from the top face down to the top of the small ball. |
| Depth from top face to top of large ball | mm | Depth reading to the top of the large ball (negative if it stands above the face). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Small ball diameter
- What it is
- The measured diameter of the smaller precision ball, which drops deeper into the internal taper.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure the ball with a micrometer rather than trusting its nominal size.
What this means: Large ball diameter
- What it is
- The measured diameter of the larger precision ball, which sits higher in the taper.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must be larger than the small ball, and both must touch the cone wall, not an edge.
What this means: Depth from top face to top of small ball
- What it is
- The depth from the top face of the part down to the top of the small ball as it rests in the taper, read with a depth gauge or height gauge.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger difference between the two depths means a smaller taper angle.
- Get it right
- Measure to the top of the ball, not its centre.
What this means: Depth from top face to top of large ball
- What it is
- The depth from the top face of the part down to the top of the large ball as it rests in the taper.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger difference between the two depths means a smaller taper angle.
- Get it right
- Enter a negative value if the large ball stands above the top face.
Method and formulas
- With radii R₁ < R₂ and H = h₁ − h₂: sin(α/2) = (R₂ − R₁) ÷ (H − (R₂ − R₁)).
- Taper on diameter = 2 tan(α/2).
- Large ball centre lies R₂ ÷ sin(α/2) from the apex; diameter at the face = 2 tan(α/2) × (that distance + R₂ + h₂).
- Example: the default readings give an included angle of 23.074° and 28.577 mm at the top face.
Limitations
- Straight, round taper; both balls must touch the conical wall all round.
- Ball diameter and depth errors are magnified for small angles.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Inspect tapered bores, valve seats and Morse sockets
- Measure a taper without a taper gauge
- Check the diameter at the gauge line
Who it is for
- Quality inspectors
- Toolmakers
- Metrology technicians
What you get
- Included taper angle
- Taper per unit length on diameter
- Diameter at the top face
Interpreting the result
- Use balls as far apart in size as the taper allows to reduce error.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 682–691. Topic reference; no source tables or recommended settings are reproduced.
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