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

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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.

Contributed by: Daisuke Ikeda, Wataru Ogasa, and Ryohei Miyadera (March 2011)
Open content licensed under CC BY-NC-SA



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].


[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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