Solids Circumscribing Spheres

1. INTRODUCTION. Although it is well known that every tetrahedron circum-scribes a sphere, the following two simple consequences apparently have not been previously recorded. First, any plane through the center of the inscribed sphere divides the tetrahedron into two smaller solids whose surface areas are equal if and only if their volumes are equal. Second, the centroid of the boundary surface of a tetrahedron and the centroid of its volume are always collinear with the center of the inscribed sphere, at distances in the ratio 4:3 from the center. This paper shows that both these and deeper results hold, not only for the tetrahe-dron or any polyhedron that circumscribes a sphere, but for more general solids called circumsolids (defined in section 4), whose faces can be curved as well as planar. The curved faces can be cylindrical, conical, or spherical. Each circumsolid circumscribes a sphere (its insphere), and all share the following property, proved in section 4: