# Paper bag problem

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On volume enclosed by two rectangles

A cushion filled with stuffing

In [geometry](/source/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](/source/Cushion) or [pillow](/source/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]

V = w 3 ( h / ( π w ) − 0.142 ( 1 − 10 ( − h / w ) ) ) , {\displaystyle V=w^{3}\left(h/\left(\pi w\right)-0.142\left(1-10^{\left(-h/w\right)}\right)\right),}

where *w* is the width of the bag (the shorter dimension), *h* is the height (the longer dimension), and *V* is the maximum volume. The approximation ignores the crimping round the equator of the bag.

A very rough approximation to the capacity of a bag that is open at one edge is:[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

V = w 3 ( h / ( π w ) − 0.071 ( 1 − 10 ( − 2 h / w ) ) ) {\displaystyle V=w^{3}\left(h/\left(\pi w\right)-0.071\left(1-10^{\left(-2h/w\right)}\right)\right)}

(This latter formula assumes that the corners at the bottom of the bag are linked by a single edge, and that the base of the bag is not a more complex shape such as a [lens](/source/Lens_(geometry))).[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

## The square teabag

A numerical simulation of an inflated teabag (with crimping smoothed out)

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

V = 1 π − 0.142 ⋅ 0.9 {\displaystyle V={\frac {1}{\pi }}-0.142\cdot 0.9}

or roughly 0.19. According to Andrew Kepert, a lecturer in mathematics at the [University of Newcastle, Australia](/source/University_of_Newcastle%2C_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](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

Robin also found a more complicated formula for the general paper bag,[1][*[specify](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources)*] which gives 0.2017, below the bounds given by Kepert (i.e., 0.2055+ ≤ maximum volume ≤ 0.217+).

## See also

- [Biscornu](/source/Biscornu), a shape formed by attaching two squares in a different way, with the corner of one at the midpoint of the other

- [Mylar balloon (geometry)](/source/Mylar_balloon_(geometry))

## Notes

1. ^ [***a***](#cite_ref-FOOTNOTERobin2004_1-0) [***b***](#cite_ref-FOOTNOTERobin2004_1-1) [Robin 2004](#CITEREFRobin2004).

## References

- Robin, Anthony C (2004). "Paper Bag Problem". *[Mathematics Today](/source/Institute_of_Mathematics_and_its_Applications#Mathematics_Today)*. **June**. [Institute of Mathematics and its Applications](/source/Institute_of_Mathematics_and_its_Applications): 104–107. [ISSN](/source/ISSN_(identifier)) [1361-2042](https://search.worldcat.org/issn/1361-2042).

- [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Paper Bag"](https://web.archive.org/web/20110629050124/https://mathworld.wolfram.com/PaperBag.html). *[MathWorld](/source/MathWorld)*. Archived from [the original](https://mathworld.wolfram.com/PaperBag.html) on 2011-06-29.[*[circular reference](https://en.wikipedia.org/wiki/Wikipedia:Verifiability#Wikipedia_and_sources_that_mirror_or_use_it)*]

## External links

- [The original statement of the teabag problem](http://www.ics.uci.edu/~eppstein/junkyard/teabag.html)

- [Andrew Kepert's work on the teabag problem (mirror)](https://web.archive.org/web/20050404002125/http://frey.newcastle.edu.au/~andrew/teabag/)

- [Curved folds for the teabag problem](https://web.archive.org/web/20050616180606/http://frey.newcastle.edu.au/~andrew/teabag/folding/curvedFold.html)

- [A numerical approach to the teabag problem by Andreas Gammel](https://web.archive.org/web/20050410213151/http://www.dse.nl/%7Eandreas/teabag.html)

- [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Paper Bag Surface"](https://mathworld.wolfram.com/PaperBagSurface.html). *[MathWorld](/source/MathWorld)*.

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Adapted from the Wikipedia article [Paper bag problem](https://en.wikipedia.org/wiki/Paper_bag_problem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Paper_bag_problem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.