Permit me to supplement Andrey's definitive answer. First, as Gerry Myerson says, this problem is discussed in Robert Lang's _[Origami Design Secrets][1]_: pp.315-318, under the title "The Margulis Napkin Problem." He credits the problem to Gregori Margulis. Second, the problem is discussed in Igok Pak's book _[Lectures on Discrete and Polyhedral Geometry][2]_, p.354, which is available online. You can pretty much guess the proof from the following instructive figure of Igor's: <br /> ![Napkin][3]<br /> Third, there is another surprising result that is intellectually analogous to increasing the perimeter by folding: The volume enclosed by any convex polyhedron can be increased by bending the surface (retaining intrinsic isometry) to render it nonconvex! See Chapter 39 of Igor's book, p.339ff. [1]: http://173-14-177-170-newengland.hfc.comcastbusiness.net/product.asp?ProdCode=1942 [2]: http://www.math.ucla.edu/~pak/book.htm [3]: https://people.csail.mit.edu/~orourke/MathOverflow/NapkinFolding.jpg