Personally, I`ve never gone the router way with making a circle.
The way I got taught (making antique oak tabletops) was to put a spike in the middle (we clamped the two halves together with a nail in the middle) and put that mess on a table saw with the nail at the radius that we wanted and pound the nail into the board that`s clamped to the saw. Put the blade all the way down to where it`s just clear and then fire the saw up. Start rotating the top and raise the blade gradually as you turn. Rotate one turn and raise the blade a bit and soon you are all the way through. A helper to do the job and hold the piece on the other side is always a good one as there are times when you just want to move your hands.
Caveats: Make sure that you have a good sturdy pin to rotate around, make sure that you keep that pin in it`s various holes and don`t let the wood kick back or it will give you a good belly punch. Also, don`t use on a 25 horse 18" table saw or you may really get bellypunched if you let it go free. The little 10" saws won`t hurt you but the big ones will let you know that you did a stupid. Also, if you keep the piece of work turning, it won`t think of coming loose.
Last but not least, know where the blade is going to come up through the wood and don`t put your hand there. The outside won`t kick anywhere as it will lift without any problems if the blade wants to move it somewhere.
It took about 5 turns to do a 3/4" oak table top 4` in diameter all the way through and only a few moments in doing it.
The router method looks about as easy to do and I`d do it in one pass with the router pointing up and just pushing the circle around with it. I`m not to sure at the moment which way is the right way to push around but that`s only a few inches to find the right direction as you want the work to kick back rather than dig in when held loosely.