spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron {p,q} is a regular q-gon. The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4 steradians) divided by the number of faces. This is equal to the angular deficiency of its dual. The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in steradians. The constant φ = is the golden ratio. Radii, area, and volume Another virtue of regularity is that the Platonic solids all possess three concentric spheres: the circumscribed sphere that passes through all the vertices, the midsphere that is tangent to each edge at the midpoint of the edge, and the inscribed sphere that is tangent to each face at the center of the face. The radii of these spheres are called the circumradius, the midradius, and the inradius. These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradius R and the inradius r of the solid {p, q} with edge length a are given by where θ is the dihedral angle. The midradius ρ is given by where h is the quantity used above in the definition of the dihedral angle (h = 4, 6, 6, 10, or 10). The ratio of the circumradius to the inradius is symmetric in p and q: The surface area, A, of a Platonic solid {p, q} is easily computed as area of a regular p-gon times the number of faces F. This is: The volume is computed as F times the volume of the pyramid whose base is a regular p-gon and whose height is the inradius r. That is, The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length, a, to be equal to 2. The constants φ and ξ in the above are given by Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest