A round bed is an unusual, but by no means unique, piece of furniture. It is rare enough that it's often easier to just build a frame for a round mattress rather than find and buy one in a shop. If you have the tools, this project should take no more than two hours unless you paint or stain the wood. If you do paint or stain, you should let the frame dry overnight.
Set your mattress on top of the plywood sheet so that no part of the mattress overlaps the plywood.
Use a pencil to trace a circle on the plywood around the outline of your mattress.
Cut the circle out of your plywood along the line you've drawn.
Cut your 1 inch by 12 inch planks so that they are each three-fourths of the diameter of the circle.
Sand one face of the plywood. You can sand the other faces if you like, but this will be the only face that's exposed at the end of construction.
Paint or finish the lumber at this stage, if you wish to paint or finish the frame.
Connect two 1 inch by 12 inch planks in an L shape by screwing them both into one of your 2 inch by 4 inch beams. The beam should rest in the inside corner of the L, with its edges flush with the edges of the planks. Use two screws per plank, drilled through the plank and into the beam.
Repeat Step 7 with the other planks and one beam.
Connect the two Ls using the remaining beams. Screw them in place as you did in Steps 7 and 8.
Position the frame of planks you've built in the center of the underside of your plywood.
Trace an outline of the end of each beam with your pencil.
Drill pilot holes in the center of each beam outline on the underside of your plywood. Drill similar pilot holes an inch deep in the center of the end of each beam. Use a drill bit an eighth of an inch narrower than the diameter of your wood screws.
Position the plywood on top of the frame, as it will lie when it's in use.
Drill a screw down through each of the pilot holes, attaching the plywood to the frame.