A regular hexagonal prism can translate and rotate about its axis inside a circular cylindrical bore. If A is its across-flats dimension, its corner-envelope diameter is D = 2A/√3. A matching hexagonal bore would prevent continuous rotation.
Regular Hexagonal Prism in Circular Bore · Ideal Envelope
Explore the circular corner envelope of a regular hexagonal prism and its independent axial motions.
Equation Used
- The prism cross-section is a centered regular hexagon with selected across-flats dimension A.
- The bore is circular and coaxial with the prism.
- Reported bore diameter is the zero-clearance corner envelope, not a production fit.
- The two views use independent normalized drawing scales.
- Clearance, tolerances, contact, torque, side load, friction, wear, strength and safety are excluded.
A practical running fit needs clearance and tolerances selected for the materials and operating conditions. The selected engaged length is a geometric input, not a minimum load-bearing recommendation.
Two independent axial motions
The cross-section shows a regular hexagon rotating within its circular corner envelope. The longitudinal view shows the prism translating through a cylindrical housing while retaining the selected engaged housing length.
What the calculator determines
Select regular-hexagon across-flats dimension and engaged length. The calculator reports the ideal minimum corner-envelope bore diameter, exact hexagonal section area, selected engagement ratio and the two ideal cylindrical-pair degrees of freedom.
Regular-hexagon envelope
For apothem a = A/2, circumradius R = a/cos30° = A/√3. Therefore D = 2A/√3 and section area = √3A²/2.
Selected-input arithmetic
For A = 25 mm, the zero-clearance corner-envelope diameter is about 28.87 mm and the regular-hexagon section area is about 541.27 mm².
Ideal geometry, not a fit specification
D is a zero-clearance geometric envelope, not a production bore size. Real design requires positive clearance and tolerances chosen for material, temperature, lubrication, contamination, speed and load. The model does not predict contact, torque capacity, bending, wear, backlash, strength or safety.
Questions
Can the hexagon rotate in a matching hexagonal bore?
No. Matching flats key the parts and leave axial translation as the intended relative motion.
Does the minimum diameter define a manufacturable fit?
No. It is the exact corner envelope only; zero clearance would not be a practical running fit.
References & Further Reading
- Wikipedia contributors. Kinematic pair. Wikipedia
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