We present a new algorithm for computing the topology of a real algebraic surface $S$ in a ball $B$, even in singular cases. We use algorithms for 2D and 3D algebraic curves and show how one can compute a topological complex equivalent to $S$, and even a simplicial complex isotopic to $S$ by exploiting properties of the contour curve of $S$. The correctness proof of the algorithm is based on results from stratification theory. We construct an explicit Whitney stratification of $S$, by resultant computation. Using Thom's isotopy lemma, we show how to deduce the topology of $S$ from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectiveness issues conclude the paper.