The paper bag problem is naturally generalized to ask the maximum possible inflated volume of the “dihedron” of any given connected planar domain obtained by gluing together its two copies along the boundaries. In this paper, given a general convex polygon satisfying a certain condition, we solve the variational problem for maximizing the volume of the flanged origami package folded from its dihedron with curved creases by the rotational origami method. We can apply our result to rectangular dihedrons to obtain origami packages with larger volume than expected by Robin’s formula. As a by-product, we also obtain large-volume pillow boxes.