With A Router
If you want to use your router a very simple method would be to make a trammel arm
. It's nothing more than a piece of ¼" plywood the width of your router base and any length (longer than the radius needed). Remove the bottom baseplate on the router, and use the machine screws to attach the plywood. You can use an ordinary straight faced carbide tipped router bit in the router. With the bit in the router, draw a straight line down the middle of the plywood.
Measure from the edge of the flute out to the dimension of the radius. For a 48" circle, measure 24". Make a cross mark on the line, and drive a finish nail down through the plywood. On your stock to be cut, find the center, and nail the arm to the center point. Place a substrate of some sort under the piece to be cut so the router bit can cut all the way through.
With the bit recessed above the work, turn on the router, and lower it into the stock about a ¼". If you have a plunge...use that function. Make a pass around the stock. Your next pass is made with the bit lowered another ¼". You do this until the piece is cut through.
With A Jig Saw
Draw your circle and cut just outside the line. When done use a block sander and smooth to a circle.
With A Table Saw
Cut your stock 48"x48". Turn it upside down and draw lines corner to corner. Where they intersect, punch a hole. Place a ¾" substrate smaller than the 48" square, and do the cross lines and drive in a finish nail into the center, and clip off the head, leaving about a ¼" of nail and file to a point.
Across from the center of the blade, attach the substrate to the table, with the nail 24" from the edge of the saw blade teeth. Lower the blade to the table. Place the piece on top of the nail and press down. Turn on the saw, and slowly raise the blade until it penetrates the stock about ⅛". Make a slow smooth pass, by turning the subject piece clockwise to the blade. If you are a lefty, and prefer to do this on the left side of the blade, you will be turning the stock counterclockwise. Your next cut is done by raising the blade another ⅛" and making a pass. This is done until the complete circle is cut.
On The Bandsaw
For a 48" circle, you need to arrange a large extension table to the right of the blade. The principle is similar to the piece with a nail. The large table should be smaller than 48", but the substrate with the nail will be fixed 24" from the right side of the blade. With this method, your starting stock should be 48"x48". Make a single pass.