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 geeral 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. We derive results for level sets of some moving average processes and for the boolean model with non-connected polyrectangular grains in R2. Applications to excursions of smooth bivariate random fields are derived in the companion paper, and applied for instance to C1,1 Gaussian fields, generalising standard results.