Summary: From previously established results in our paper [Ramanujan J. 5, No. 1, 73‒89 (2001; Zbl 0991.42004)], we develop a simple proof of {\it K. Ball}’s expression in [Proc. Am. Math. Soc. 97, 465‒473 (1986; Zbl 0601.52005)] for the volume of the intersection of an $(n-1)$-dimensional hyperplane with an $n$-dimensional cube, as well as a simple proof of the formula given by {\it R. Frank} and {\it H. Riede} [Am. Math. Mon. 119, No. 10, 868‒872 (2012; Zbl 1264.51015)] for that volume.