Top view of the secondary cage, where the trapezium shape of the 8 walls can be seen. They overlap at the joints, so only 45° angles needed to be machined in the wood, using a 45° router bit. The walls were made slightly oversized when glued onto the rings, and sanded down later to cut back the excess sticking out.
The whole construction is made of 9mm birch plywood. The inner diameter of the rings is 274mm, leaving 9mm for the thinnest parts. Adding 9mm for the walls, the total cage diameter becomes 310mm.
The octagonal rings can be trued before cutting out the center, by drilling a center hole (that's needed lateron for the router jig anyway) and screwing the two plates together. Rotating 45° with respect to eachother should give a good match each time, sand down if not.