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If I understand the question properly, the answer depends on the flexibility of the sheet. A fine cloth napkin might crease almost like origami, in which case the height would be half the diagonal of the square, $\sqrt{2}/2$ for a unit square:
Square fold http://cs.smith.edu/%7Eorourke/MathOverflow/OctFold.jpg
For more realistic physical models, the shape would be more bag-like, and the analysis more difficult.