Given the equations for the n + 1 hyperplanes that bound an n-simplex in Rn, simple formulae are derived for the contents of the n — r simplices (0 ≤ r ≤ n) embedded in it. For example, when n = 3, the formulae include the volume of the tetrahedron, the areas of its faces and the lengths of its edges. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 14 N 99.