  It has been found in numerical experiments that when one removes a sector
from an elastic sheet and glues the edges of the sector back together, the
resulting configuration is radially symmetric and nearly conical. We make a
rigorous analysis of this setting under two simplyfying assumptions: Firstly,
we only consider radially symmetric configurations. Secondly, we consider the
so-called von-K\'arm\'an limit, where the size of the removed region as well as
the deformations are small. We choose free boundary conditions for a sheet of
infinite size. We show existence of minimizers of the suitably renormalized
free energy functional. As a by-product, we obtain a lower bound for the
elastic energy that has been conjectured in the related context of d-cones.
Moreover, we determine the shape of minimizers at infinity up to exponentially
decaying terms.
