Formulae are presented that
give the content of a simplex in Euclidean n-space: (i) in terms of the lengths of and
the angles between the vectors from a fixed point to the vertices of the simplex; (ii)
in terms of the lengths of and the angles between the perpendiculars from a
fixed point to the bounding faces of the simplex. We then determine the
largest simplex whose vertices are given distances from a fixed point and we
determine the smallest simplex whose faces are given distances from a fixed
point. As special cases we find that the regular simplex is the largest simplex
contained in a given sphere and is also the smallest simplex containing a given
sphere.
