Maximizing the Volume of a Cup Made from a Square Sheet of Paper

This Demonstration studies the maximization of the volume of a cup shaped like a truncated square pyramid made from a square sheet of paper of size 12×12 cm. This may be a new kind of problem in Origami.

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DETAILS

The volume of this 3D figure is . We look for the maximum value of .
By and we have
and .
From these two equations we get and .
Using Mathematica to factor, .
Again using Mathematica to solve, it is easy to see that we get the maximum value when and , and the maximum value is .
According to the referee of the Journal Origami Society of Japan, this is a new problem proposed by the authors.
The origami wrapping problem, which has been studied by some mathematicians, looks for the biggest object that can be wrapped with a sheet of paper [1].
Reference
[1] M. L. Catalano-Johnson and D. Loeb, "Problem 10716: A Cubical Gift," American Mathematical Monthly, 108(1), 2001 pp. 81–82.

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Related Curriculum Standards

US Common Core State Standards, Mathematics