Simple way to build large circular plates?
The largest truely round base plates in the LEGO system fit in a 8x8 square. You can build them from two LegoPlate, Round Half 4 x 8 or from four quarters.

For a bigger (and thicker!) base plate there is a Duplo version of the above mentioned base plate, fitting into a 16x16 studs LEGO system square: LegoDuplo, Plate Round Corner 4 x 8 Double

EDIT: For a thicker base, you can use these bricks: Brick, Round Corner 10 x 10 with Slope 33 Edge. They make a base fitting into a 20x20 square.