We establish an expression of the Euler characteristic of a r- regular planar set in function of some variographic quantities. The usual C2 framework is relaxed to a C1,1 regularity assumption, generalising existing local formulas for the Euler characteristic. We give also general bounds on the number of connected components of a measurable set of R2 in terms of local quantities. These results are then combined to yield a new expression of the mean Euler characteristic of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. Applications to excursions of bivariate random fields are derived in the companion paper, and applied for instance to C1,1 Gaussian fields, generalising standard results.