Paper bag problem

In geometry, the paper bag problem or teabag problem is to calculate the maximum possible inflated volume of an initially flat sealed rectangular bag which has the same shape as a cushion or pillow, made out of two pieces of material which can bend but not stretch.

According to Anthony C. Robin, an approximate formula for the capacity of a sealed expanded bag is:[1] where w is the width of the bag (the shorter dimension), h is the height (the longer dimension), and V is the maximum volume.

[citation needed] For the special case where the bag is sealed on all edges and is square with unit sides, h = w = 1, the first formula estimates a volume of roughly or roughly 0.19.

According to Andrew Kepert, a lecturer in mathematics at the University of Newcastle, Australia, an upper bound for this version of the teabag problem is 0.217+, and he has made a construction that appears to give a volume of 0.2055+.

[citation needed] Robin also found a more complicated formula for the general paper bag,[1][specify] which gives 0.2017, below the bounds given by Kepert (i.e., 0.2055+ ≤ maximum volume ≤ 0.217+).