||This paper addresses the approximation of belief functions by probability functions where theapproximation is based on minimizing the Euclidean distance. First of all, we simplify this optimization problem so itbecomes equivalent to a standard problem in linear algebra. For the simplified optimization problem, we provide theanalytic solution. Furthermore, we show that for Dempster-Shafer beliefthe simplified optimization problem is equivalent to the original one.In terms of semantics, we compare the approximation of belief functions to various alternative approaches, e.g.pignistic transformation for Dempster-Shafer belief and Shapley value for fuzzy belief functions. For the later one,we give an example where the approximation method has someobvious statistical advantages.Additionally, for the approximation of additive belief functions, we can provide a semantical justification.