Step 7: Create wire frame
Make three wire triangles the size of the tetrahedron sides and duct tape them together into a pyramid. Make absolutely certain that all sharp edges are firmly duct taped in place so that there are no unpleasantly pokey parts to injure your cat. Forming the pyramid out of three finished triangles has the benefit of creating nice smooth pieces, and also having two wires along each edge will give it extra support because coat hanger wire isn't that strong.
When the frame is ready simply pull the cloth over it and use safety pins or basting stitches to hold it in place. You don't want to permanently attach the frame because you may want to wash the costume in the future. All that you need is something to hold the bottom edge of the cover to the bottom edge of the frame, and safety pins hidden on the inside work perfectly for that.