Question 92
Let $D$ be a dodecahedron which can be inscribed in a sphere with radius $R$. Let $I$ be an icosahedron which can also be inscribed in a sphere of radius $R$. Which has the greater volume, and why?
Note: A regular polyhedron is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with 12 faces which are regular pentagons and a regular icosahedron is a polyhedron with 20 faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere.
The illustration below shows a dodecahedron and an icosahedron, not necessarily to scale.
Note: A regular polyhedron is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with 12 faces which are regular pentagons and a regular icosahedron is a polyhedron with 20 faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere.
The illustration below shows a dodecahedron and an icosahedron, not necessarily to scale.