On the Diameter of a Circle to Hold a Cube - Maehara, Hiroshi (Computational Geometry, Graphs and Applications, 2011)
It is proved that a circle of diameter d can hold a cube of unit edge if and only if $\sqrt2\le d< 1.53477...$ . Similarly, it can be proved that a circle of diameter d can hold a regular octahedron of unit edge if and only if 1 ≤ d < 1.1066...