# How do I find the maximum volume of an A4 piece of paper using the isoperimetric inequality?

Through my research on this site, I recently stumbled across a post where it was stated that the maximum volume for a sheet of A4 paper (210mm x 297mm) is < 2.072 l. This was found to be so using the isoperimetric inequality; I was wondering how this person possibly conducted their working, and if it was correct, as I need to find the maximum volume of an A4 piece of paper (which can be cut and pasted). The shape will certainly be a sphere, but I need to know how the author of this question (linked below) came to his conclusion. I do not yet have enough "reputation" to comment, so I appreciate all help, as a mathematical novice.

In short, I would appreciate someone telling me how to find an isoperimetric inequality (with complete working) of an A4 sheet of paper, and if the asker of the original question was correct in his inequality. Please do not hesitate to ask clariying questions.

Original question: What is the maximum volume that can be contained by a sheet of paper? Asker states isoperimetric inequality in Clarifications section of question.

## 1 Answer

The isoperimetric inequality says that the largest volume enclosed by a surface of given area is the volume of the sphere of this area. In the linked problem we do not want to "enclose" fluid, but just "hold" it. One then has to prove that the largest volume that can be "hold" in a surface of given area is the volume of a half ball whose spherical boundary part has the given area. Given that we can argue as follows:

The A4 sheet has area $$A=210\cdot297$$ mm$$^2$$ If we cut up this sheet into tiny strips and glue the strips together to a half sphere of radius $$r$$ we obtain $$2\pi r^2, or $$r<99.63$$ mm. The volume of the resulting half ball then is $${2\pi\over3}r^3<2.071$$ liter.