inscribed in a sphere or a cube

Tetrahedron |
Hexahedron |
Octahedron |

Dodecahedron |
Icosahedron |

In book XIII of his Elements
Euclid discusses the construction of the regular polyhedra inscribed in a given sphere.
Each proposition reads as

* stands for one of the polyhedra and ** expresses a relation between the diameter of the given sphere and the side (the edge) of the polyhedron.

For four polyhedra Euclid starts with the given sphere, or with its diameter and gives a construction of the side of the polyhedron.

As to the dodecahedron he starts with a cube, constructs a dodecahedron and finds the circumscribing sphere. This construction is explained in Golden section - Pentagon - Dodecahedron.

However, it looks easier to construct each of the regular polyhedra starting from a given cube. For the tetrahedron, the cube (of course) and the octahedron, the images clearly indicate how to do it. For those three polyhedra it's not difficult to calculate the radius of the circumscribing sphere.

More about the icosahedron and the dodecahedron: Dodecahedron and Icosahedron.

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