Smallest square to wrap a cylinder Suppose you need to gift-wrap a cylinder (e.g., a can of tennis balls, or a large candle)
of height $h$ and radius $r$.
Here wrap is the natural sense of covering the surface area of the cylinder completely,
without cutting the square, creasing however needed.
What is the smallest square that suffices for a given $h$ and $r$?
For example, a rectangle of dimensions $(h+2r) \times (2 \pi r)$ suffices
for how one might wrap a can of tennis balls or a stout candle:

          

   


In this $h=3$ and $r=1$ case, the rectangle has dimensions $5 \times 6.28$, 
and so a square of side $2 \pi$ suffices.  But is that optimal?
          Merry Christmas!
 A: Perhaps Joseph can fill in this description with a picture.
In square with vertices A, B, C, and D in clockwise order, draw a line
from A to a point P on BC.  If the length of AP is longer than the
circumference R of the inscribed circle of ABP, then a cylinder with
circumference R and height which I leave you to determine can be
wrapped by ABCD.

          


                              (Image added by J.O'Rourke)

Gerhard "Wishing You A Happy New Year" Paseman, 2012.12.25
A: You can wrap the cylinder with a square whose diagonal has length $2(h + 2r)$: put the cylinder in the center of the square and fold the corners up to meet at the top of the cylinder. When $h/r$ is less than $\pi - 2$, this method uses less paper than the OP's method. If $h/r$ is very small (if you're wrapping a CD, for example), it uses just a little over $2/\pi$ as much paper as the OP's method.
I haven't done a comparison with Gerhard Paseman's method...
A: This is in response to Vectornaut's question about wrapping with a thin rectangle.
In Geometric Folding Algorithms: Linkages, Origami, Polyhedra, 
it is argued (Theorem 15.2.1, p.236) that any polyhedron can be covered with a thin
strip with arbitrarily
small surface area beyond that of the polyhedron.
A polyhedral approximation to a cylinder then yields the claim.
This is Figure 52.2 (p.234), which gives some idea of switchback turns of the strip
used in the argument:

          


Here is a link to the original 1999 paper by Demaine, Demaine, and Mitchell,
"Folding Flat Silhouettes and Wrapping Polyhedral Packages: New Results in Computational Origami": link.
