Barrel / Cask Volume (Bulged Sides)

The volume of a barrel or cask that swells to a larger middle (bung) diameter D than its ends (head diameter d) over a length L. Two standard closed forms bracket it: parabolic staves (exact for a parabolic profile) V = (pi L/15)(2 D^2 + D d + 3/4 d^2), and circular-arc staves (Kepler's classic rule) V = (pi L/12)(2 D^2 + d^2); a real barrel sits between them, here under 1% apart. Both collapse exactly to the straight cylinder pi(D/2)^2 L when D = d, so a nearly straight drum reads like a cylinder. A 27 in bung, 24 in head, 36 in long barrel holds 82.8 gal (parabolic) to 83.0 gal (circular). Enter inside dimensions in inches. The staved geometry is idealized; a strapping chart or a water fill governs custody. US gallons (231 in^3).

Parabolic staves V = (pi L/15)(2D^2 + Dd + 3/4 d^2) (exact for a parabolic profile); circular-arc staves V = (pi L/12)(2D^2 + d^2) (Kepler); both collapse to the cylinder when D = d.

The barrel / cask volume - standard solid geometry (Kepler's barrel rule) as in Machinery's Handbook (Industrial Press), by name; public domain.

Pure solid geometry, public; the bung diameter, head diameter, and length are user-supplied.

Estimate. AHJ and licensed professional govern.

Field names used by the API: bung_diameter_in, head_diameter_in, length_in, parabolic_gal, circular_gal

• Parabolic staves V = (pi L/15)(2D^2 + Dd + 3/4 d^2), exact for a parabolic profilesolid geometry (integration)
• Circular staves V = (pi L/12)(2D^2 + d^2), Kepler's barrel ruleKepler; Machinery's Handbook
• Scope bung diameter >= head diameter (bulges outward); both forms collapse to the cylinder when D = dscope of this tile