 For any subset formula_4 with exactly "k" elements, let formula_5 be the orthogonal projection of "U" onto the linear span of formula_6, where formula_7 and formula_8 is the standard basis for formula_2.Then

where formula_11 is the "k"-dimensional volume of "U" and the sum is over all subsets formula_4 with exactly "k" elements.De Gua's theorem and its generalisation (above) to "n"-simplices with right-angle corners correspond to the special case where "k" = "n"−1 and "U" is an ("n"−1)-simplex in formula_2 with vertices on the co-ordinate axes.