A polymer film draping over a point of contact will wrinkle due to the strain imposed by the underlying substrate. The wrinkle wavelength is dictated by a balance of material properties and geometry; most directly the thickness of the draping film. At a critical strain, the stress in the film will localize, causing hundreds of wrinkles to collapse into several discrete folds. In this Letter, we examine the deformation of an axisymmetric sheet and quantify the force required to generate a fold. We observe that the energy of formation for a single fold scales nearly linearly with the film thickness. The onset of folding, in terms of a critical force or displacement, scales as the thickness to the four-ninth power, which we predict from the energy balance of the system. The folds increase the tension in the remainder of the film causing the radial stress to increase, thereby decreasing the wavelength of the remaining wrinkles.