A polishing system includes a platen having a top surface, an annular polishing pad supported on the platen, a carrier head to hold a substrate in contact with the annular polishing pad, a support structure from which the carrier head is suspended and which is configured to move the hold the carrier head laterally across the polishing pad, and a controller. The platen is rotatable about an axis of rotation that passes through approximately the center of the platen, and the inner edge of the annular polishing pad is positioned around the axis of rotation. The controller is configured to cause the support structure to position the carrier head such that a portion of the substrate overhangs the inner edge of the annular polishing pad while the substrate is contacting the polishing pad.