Make a sort of compass that guides a router instead of pencil.

Screw a strip of wood a few inches wide by ~2 feet long to the base of your router, with the router at one end of the board. Drill your shaft hole in the plywood, and drill the same size hole in the board attached to the router, with the hole centered 18" from the inside of the bit. Using a dowel that matches your shaft size as a center pivot, you can now swing the router in a neat circle around the center bore.

Of course this is only economical if you already have (or plan to buy) a router. If you don't, you can use some basic geometric construction techniques to find the center of an existing circle

A jig on a bandsaw is the best way to go.
Casey

It would be helpful to know which tools you have already. Either of the above would work, and there are a few more.