Measure the floor of your existing stage, and determine the center point. For a rectangular stage, extend a diagonal line from each front corner to the back corner on the opposite side of the stage. The lines cross at the center point, which is the pivot point for your rotating stage.
Screw a flange into the floor at the center point. Use a flange that is not threaded inside. Tighten the flange down well, as your entire stage will pivot around this flange. Using 1/4-inch wood screws that are 1 inch long, place the screws in the pre-drilled holes on the flange. Hold the flange in position as you screw the screws through the holes in the flange into the floor.
Cut a 6-inch length of pipe of a diameter that fits into the flange. Push the pipe into the flange. Screw the set screw in the side of the flange down tightly so that the pipe does not move.
Join plywood sheets together to make your rotating stage the size you want it. Though the sheets are square, join them before cutting the circle you will need for your rotating stage. Join 4-by-8-foot sheets together using butt joints with blocks. Butt the edges of the sheets together, and glue them. Cut 6-by-18-inch blocks from plywood, and screw them to the undersides of the joints, using 3/4-inch screws so they won't stick out. Place each block so that it straddles the joint. Space the blocks along the length of the joint, leaving 6 inches between blocks.
Saw a circle out of the joined plywood. This will be the floor of your rotating stage. If the cut intersects a place where you have supported a joint with a block, continue the cut. You will have a partial block support at this point, but it will be strong enough to support your actors, because the next block will be no further than 6 inches away.
Screw a flange that is the same size as the flange on the old stage floor into the center of the rotating platform, using screws that do not penetrate to the other side of the floor. Note that at this point your rotating stage floor is upside down to make it easier for you to work on it.
Screw straight-line castors to the edges of the rotating stage floor. Space them out to support the edges of your rotating stage floor. Place the castors 2 feet apart so that the edge of the stage will be adequately supported.
Add an additional circle of castors halfway between the edge of the platform and the center point. Place these a foot apart to add strength to the platform.
Turn your entire rotating stage platform over, and place the flange on top of the pipe that is secured in the flange in the stage floor. Do not tighten the set screw in the second flange, because you want the flange to rotate around the pipe. The casters are now on the stage floor holding up the edges of your rotating platform.
Attach a black rope to the edge of the rotating platform by screwing a large eye screw into the platform and tying the rope to it. This allows you to pull the rotating platform, making it rotate.