  Let $D$ be a Riemannian 2-disc of area $A$, diameter $d$ and length of the
boundary $L$. We prove that it is possible to contract the boundary of $D$
through curves of length $\leq L + 200d\max\{1,\ln {\sqrt{A}\over d} \}$. This
answers a twenty-year old question of S. Frankel and M. Katz, a version of
which was asked earlier by M.Gromov.
  We also prove that a Riemannian $2$-sphere $M$ of diameter $d$ and area $A$
can be swept out by loops based at any prescribed point $p\in M$ of length
$\leq 200 d\max\{1,\ln{\sqrt{A}\over d} \}$. This estimate is optimal up to a
constant factor. In addition, we provide much better (and nearly optimal)
estimates for these problems in the case, when $A<<d^2$. Finally, we describe
the applications of our estimates for study of lengths of various geodesics
between a fixed pair of points on "thin" Riemannian $2$-spheres.
