How can you compute the maximum volume of an envelope(used to enclose a letter)? It's obvious that the volume of a envelope is 0 when flat and non-0 when you open it up. However, if you were to fill it with liquid, there must be some shape where it has a maximum volume. Is there a practical way to determine that volume?
I think a starting point would be to start with a cylinder and seal one end by putting it in something like a vice. That gives you the same shape.
EDIT: The reason this problem came to mind is because Quaker's oatmeal packets have a fill line on them so that you don't need a measuring cup. Every day, as I fill up the little packet with water, I think "what a clever idea," but also I wonder about the math behind it. I expect they approximate because, after all, it is just oatmeal.
 A: As the envelope is made of paper, a mathematical model of its deformation would be an isometric coordinate transformation and thus must have zero Gauss curvature almost everywhere; that restriction is clearly violated by the cited solution of the teabag problem.  
I already pointed out in a comment, that there was an article in the German issue (Spektrum der Wissenschaft, June 1995) of Scientific American, where the topic has been discussed here (in German).
Maybe the same topic has also been treated a few month earlier in Scientific American.
The essential ressource is Edouard Baumann, especially this link (unfortunately in German only); there is also an image of such a polyhedral cushion.
A: Your question is a variant of the teabag problem.
I don't believe an exact answer is known, but
for the $1 \times 1$ square teabag, the maximum volume is about $0.2$:

          


          

(Image from Wikipedia article.)




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*The primary reference is Anthony Robin's 2004 article, "Paper Bag Problem".
Mathematics Today—Bulletin of the Institute of Mathematics and its Applications. 40.3 (2004): 104-107.

*MathWorld article on "paper bags", which quotes Robin's 2004 volume bound.

*$13$-second YouTube Simulation of Inflating a Teabag, using a cloth simulation algorithm.
