Calculation
α = 2·atan[(D₁−D₂)/(2l)]
The half-angle α/2 describes one cone side relative to the axis; α is the full included angle.
Roloff/Matek Gl. (12.25–12.26)
Cone angle from diameters and length
The half-angle α/2 describes one cone side relative to the axis; α is the full included angle.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Included cone angle α — Full included cone angle; the setup half-angle is α/2.
- Large diameter D₁ — Larger end diameter of the cone.
- Small diameter D₂ — Smaller end diameter of the cone.
- Cone length l — Axial distance between the two diameter planes.
Example
D₁=60 mm, D₂=50 mm and l=100 mm give α≈5.72° and C=(D₁−D₂)/l=1:10.
Assumptions and limits
Straight cone of revolution; D₁>D₂ and axial length. Joint tolerances and capacity are excluded.
Understand Cone angle from diameters and length
Cone angle and taper ratio is a focused preliminary calculation based on Roloff/Matek. The calculator rearranges the closed-form relationship for every included quantity and deliberately separates this result from a complete component verification.
What does this quantity describe?
Taper is C=(D1−D2)/l. From tan(α/2)=(D1−D2)/(2l) follows the full included cone angle α; the setup angle of one cone side is α/2.
Formula and variables
α = 2·atan[(D₁−D₂)/(2l)]
C = (D1−D2)/l; tan(α/2) = (D1−D2)/(2l)
| Symbol / input | Meaning |
|---|---|
| Included cone angle α | Full included cone angle; the setup half-angle is α/2. |
| Large diameter D₁ | Larger end diameter of the cone. |
| Small diameter D₂ | Smaller end diameter of the cone. |
| Cone length l | Axial distance between the two diameter planes. |
Choose the inputs correctly
D1 is the large and D2 the small diameter. l is the axial distance between diameter planes, not slant length. The calculator returns full angle α.
How to use the calculator
Select the target, enter all remaining quantities for the actual component, and verify the units. Then compare the result with the stated model limits and with the required strength, safety and operating checks.
Worked example
60 mm, 50 mm and 100 mm give C=0.1=1:10 and α=5.72°; setup half-angle is 2.86°.
Understand the result and units
Distinguishing full angle α from setup half-angle α/2 prevents a common factor-of-two error in manufacture and inspection.
The calculator converts internally to coherent SI units. Length, force, torque, stress and angle may therefore use the offered units; dimensionless factors are entered as decimals.
What is taken from Roloff/Matek
Only the closed-form relationship from Kapitel 12.3.2, Gleichungen (12.25) und (12.26) is used. Tabulated data, material limits and detailed design checks are not silently added; they remain explicit inputs or are expressly outside the model.
Typical applications
Cone angle and taper ratio supports option comparison, plausibility checks and early sizing within its machine-element cluster. Releasing a design requires the additional checks described in the cited chapter.
Assumptions, limits and common mistakes
Straight cone of revolution with D1>D2 and axially measured length; tolerances, contact, self-locking and capacity are not part of this geometry.
Common mistake: Typical errors are misreading the effective length or force, entering percentages instead of decimals, and treating a preliminary result as a complete verification. In particular: Straight cone of revolution with D1>D2 and axially measured length; tolerances, contact, self-locking and capacity are not part of this geometry.
Frequently asked questions
Is Cone angle and taper ratio a complete strength verification?
No. Straight cone of revolution with D1>D2 and axially measured length; tolerances, contact, self-locking and capacity are not part of this geometry.
Where do the equation and validity limits come from?
Roloff/Matek, Machine Elements, Kapitel 12.3.2, Gleichungen (12.25) und (12.26); the local 21st edition was cross-checked against the available 24th edition.
Can I order the calculated value directly as a nominal size?
Only after matching it to available standard or manufacturer series and completing the additional safety, material and operating checks.