Calculations at a barrel. Geometrically, a barrel is a truncated prolate spheroid, with circles of equal size on top and on bottom. The formula for the barrel volume was discovered by Johannes Kepler. The surface area of a slightly bent barrel is close to that of a cylinder with the radius R. With a greater deviation from the cylindrical shape, the surface area per volume of the barrel increases.
Enter both radiuses and the height and choose the number of decimal places. Then click Calculate.
Formulas:
d = √ h² + ( 2 * r )²
V = π * h * ( 2R² + r²) / 3
pi:
π ≈ 3.141592653589793...
Radiuses, height and diagonal have the same unit (e.g. meter), the volume has this unit to the power of three (e.g. cubic meter).
The exact volume of a barrel can only be determined using very complicated integrals, which cannot be represented algebraically and therefore cannot be converted into a calculable formula. Johannes Kepler's approximation formula, the so-called Kepler's barrel rule, assumes that the barrel has a parabolic outer boundary, so it is better the more the side lines of the cross-section can be represented by a parabola. Johannes Kepler ordered several barrels of wine for his wedding in 1615 and was annoyed that the wine merchants measured the volume of the barrels as if they were cylindrical, so he started to study this topic.
Kepler's barrel rule can be used for every solid of revolution if their generating curves can be simulated by parabolas. The extension of Kepler's barrel rule to curves in general, especially those that are difficult to integrate, is Simpson's rule. Its application is complicated, but leads to good approximations and an error estimate. Simpson's rule was discovered by Thomas Simpson in the mid-18th century.