I assume that we're talking about an ordinary polyhedron that looks topologically like a sphere, not like some kind of torus or multi-sheeted sphere or whatnot. Apply Euler's Formula: Vertices–Edges+Faces = 2, for a polyhedron. Plugging in your numbers gives V–24+10 = 2, from which we see that V = 16. Alternative solution: Draw an octagonal prism, and confirm that your figure has 16 vertices, 10 faces, and 24 edges. Written Apr 13, 2013 · View Upvotes.