Cutting perfect circles is easy with a band saw—as long as you take a half hour to build yourself a cutting jig. I've cut circles with radiuses ranging from a few inches to a couple of feet. The only limit is the distance from the fence to the saw blade.
Use any 3/4-in. plywood to make the jig, and attach a runner to the underside that fits in the band saw's miter gauge slot. That will hold everything steady while you turn the actual circle stock through the blade. Then make yourself an auxiliary fence from another piece of 3/4-in. plywood as shown. Send the jig through the blade to create the saw kerf, then shut off the saw when the blade reaches the centerline and lock the fence about 1/8 in. away from the jig. Clamp the auxiliary fence to the band saw fence with the stop against the edge of the jig. Hold it slightly above the jig so the jig will slide easily beneath it. The auxiliary fence not only stops the jig at the right spot for turning circles but also keeps the jig from tipping up from the band saw table.
Cutting is simple. You'll need to choose narrow blades for small circles. Push the jig with the mounted cutting blank until you reach the stop and then just twist the blank through the blade. You'll have a small amount of cleanup to do where the cut starts and stops. You're done.