How to make circular rings.  I bought a threaded steel rod, an eye bolt, and four hex nuts at Home Depot for less than ten bucks.  I attached the steel rod and eye bolt to my router, allowing the router to rotate in a circular motion.  The eye bolt is adjustable allowing you to create different size circles.  (See pictures below).

Note: these pictures were taken before I cut the rings.  But you'll get the ideal.

Router assembly with threaded steel rod and eye bolt.

Close-up view of the threaded steel rod attached to the router.  The eye bolt is adjustable to whatever radius you determine.  A hole was drilled and the eye bolt inserted in the center of the circle, allowing the router to rotate in a circular motion.  Tighten the hex nuts with a wrench before cutting your circles.  Any loose connections and your circles will be out-of-round.  Also secure your mdf boad before cutting.  I secured my mdf board by screwing it down to an old speaker box.  Place at least one screw next to the eye bolt and another outside the outer ring of your circle so the mdf board doesn't  move.  And now your ready to start cutting.  Wear safety glasses and a respirator, you'll be creating a lot of dust. 

The final result will look great.  I glued these two rings together.  The edges were smooth and the outer edges were a perfect match.

Good Luck!

Hey, quick question-

What kind of router bit did you use to cut all the way through?

Good question.

I actually had to make two cuts to cut completely through the 3/4" mdf board.  After making the first cut, I had to plunge the router down approximately 1/32" to cut completely through the mdf board.  The second cut passed through the mdf board like butter.

Thank you.