If you want to control the angle that the plane moves around the cylinder then make a new sketch on the end of the cylinder draw a line from the centerpoint of the projected circle to the edge of the circle.  Exit the sketch place a new workplane on the endpoint of the line then pick the line itself.  This will make a workplane that formed normal to that line tangent to the cylinder.  now that you have a workplane make a new sketch project the top and bottom edges of the cylinder that you see when you "Look At" the sketch. Draw a line from one projected edge to the other.  Then Draw a circle from the midpoint of the line. No big deal.

You need to build tangent work plane to the cylinder.

Enter work plane command and press right mouse button on the screen. A visual help will appear and you can pick wich film to see about creating work planes.

After creating it, Crete a new sketch on it ang project the circular sketch for reference.

Hope I helped.