Intuitively, a set of sites on a surface is in Euclidean position if points are so close to each other that planar algorithms can be easily adapted in order to solve most of the classical problems in Computational Geometry. In this work we formalize a definition of the term “Euclidean position” for a relevant class of metric spaces, the Euclidean 2-orbifolds, and present methods to compute whether a set of sites has this property. We also show the relation between the convex hull of a point set in Euclidean position on a Euclidean 2-orbifold and the planar convex hull of the inverse image (via the quotient map) of the set.