Cylinder

A cylinder is a solid of circular Cross-Section in which the centers of the Circles all lie on a single Line. The cylinder was extensively studied by Archimedes in his 2-volume work On the Sphere and Cylinder in ca. 225 BC.

A cylinder is called a right cylinder if it is ``straight'' in the sense that its cross-sections lie directly on top of each other; otherwise, the cylinder is called oblique. The surface of a cylinder of height $h$ and Radius $r$ can be described parametrically by

$$x = r\cos\theta$$ (1)

$$y = r\sin\theta$$ (2)

$$z = z,$$ (3)

for $z\in [0,h]$ and $\theta\in [0,2\pi)$. These are the basis for Cylindrical Coordinates. The Surface Area (of the sides) and Volume of the cylinder of height $h$ and Radius $r$ are

$$S = 2\pi rh$$ (4)

$$V = \pi r^2h.$$ (5)

Therefore, if top and bottom caps are added, the volume-to-surface area ratio for a cylindrical container is

$${V\over S}={\pi r^2h\over 2\pi rh+2\pi r^2}={1\over 2}\left({{1\over r}+{1\over h}}\right)^{-1},$$ (6)

which is related to the Harmonic Mean of the radius $r$ and height $h$.

See also Cone, Cylinder-Sphere Intersection, Cylindrical Segment, Elliptic Cylinder, Generalized Cylinder, Sphere, Steinmetz Solid, Viviani's Curve

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 129, 1987.