  In this paper we are concerned with the problem of finding hypersurfaces of
constant curvature and prescribed boundary in the Euclidean space, using the
theory of fully nonlinear elliptic equations. We prove that if the given data
admits a suitable radial graph as a subsolution, then we can find a radial
graph with constant curvature and that realizes the prescribed boundary. As an
application we prove that if $\Omega\subset\mathbb{S}^n$ is a mean convex
domain whose closure is contained in an open hemisphere of $\mathbb{S}^n$ then,
for $0<R<n(n-1),$ there exists a radial graph of constant scalar curvature $R$
and boundary $\partial\Omega.$
