It is not difficult to see that , and we can express the volume of this 3D figure with . The volume is
where . This inequality is needed to construct the 3D figure.
We look for the maximum value of by mi=NSolve[D[v[x],x]==0,x]//Last;v[x]/.mi and we get .
Maximizing the volume of a cup made from a square sheet of paper by using rotations is a problem proposed by the authors. Our previous research is presented in . 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 (see ).
 M. L. Catalano-Johnson and D. Loeb, "Problem 10716: A Cubical Gift," American Mathematical Monthly, 108(1), 2001 pp. 81–82.
 W. Ogasa, T. Inoue, K. Nishimura, T. Nakaoka, D. Ikeda, A. Kanno, S. Nakamura, H. Matsui, T. Yamauchi, S. Utsuki, and R. Miyadera, "The Maximization of a Cup Made from a Square Sheet of Paper," Science of Origami, 1(1), 2011, in Japanese.