For each face of the tetrahedron, construct a new tetrahedron with that face as the base and the center of the sphere as the fourth vertex. Now the original tetrahedron is divided into four smaller ones, each of height R. The volume of a tetrahedron is Ah/3 where A is the area of the base and h the height; in this case h=R. Combine the four tetrahedra algebraically to find that the volume of the original tetrahedron is R/3 times its surface area.

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