# cylinder

A cylinder is a three-dimensional surface described by
the Cartesian equation (*x*/*a*)^{2} + (*y*/*b*)^{2} = 1. If *a* = *b* then the surface is a **circular cylinder**,
otherwise it is an **elliptic cylinder**.

The cylinder is a degenerate quadric because
at least one of the coordinates (in this case *z*) doesn't appear in
the equation, though by some definitions the cylinder isn't considered to
be a quadric at all. In common usage, a cylinder is taken to mean a finite
section of a right circular cylinder with its ends closed to form two circular
surfaces.

If the cylinder has a radius *r* and a length *h*, then its volume
is *V* = π*r *^{2}*h* and its surface area is *A* = 2π*r*^{ 2} + 2π*rh*. For a given volume,
the cylinder with the smallest surface area has *h* = 2*r*. For
a given surface area, the cylinder with the largest volume has *h* = 2*r*. More unusual types of cylinder include the **imaginary
elliptic cylinder**: (*x*/*a*)^{2} + (*y*/*b*)^{2} = –1, the **hyperbolic cylinder**: (*x*/*a*)^{2} – (*y*/*b*)^{2} = 1, and the **parabolic cylinder**: *x*^{2} + 2*y* = 0.