For the closed surface \(\Gamma\) in \(\mathbb{R}^ 3\) given by the local parameter representation \(f_ i: \Omega_ i\to\mathbb{R}^ 3\) with \(\Omega_ i\) open domains in \(\mathbb{R}^ 2\), \(i=1,\dots,p\), the authors construct an \(h\) parameter-dependent family \(\Gamma_ h\) of approximations to \(\Gamma\) such that \(\Gamma_ h\) is a triangulated manifold. There are also considered some refinements \(\Gamma_ h\), of the family \(\Gamma_ h\). It is presented an algorithm for the construction of the triangulations of the parameter domains and of the approximations \(\Gamma_ h\) such that no overlappings of the triangulations appear. Finally, for the approximation of smooth manifolds asymptotic error estimates are given.