Circumradius of the dodecahedron

Suppose the dodecahedron shown below has edge length 1. Consider the blue regular pentagon ABCDE and the red regular pentagon FGHIJ. The sides of each of these pentagons is a diagonal of a face of the dodecahedron. So the sides of these two pentagons have length τ, the golden ratio. Thus the diagonals AC and FI have length τ². So the rectangle ACIF has height 1 and base τ². The diagonal AI of this rectangle is a circumdiameter of the dodecahedron. Thus the circumradius of the dodecahedron is 1/2√(1 + τ⁴).