The pilot performed a preflight inspection for the cross-country flight, which included checking the fuel tanks. He used a calibrated stick and determined that he had 15 gallons in the right fuel tank and 10 gallons in the left fuel tank; he flew the majority of the flight with the right fuel tank selected. Upon reaching the destination airport, as he was switching from the right fuel tank position to the "BOTH" position, the engine surged. As the pilot made a left turn onto the base leg of the traffic pattern, he noted that the airplane was slightly high. He initiated a left slip to lose altitude. When he leveled the airplane, the engine began to surge as if it were out of fuel. The pilot reported that he tried to maintain engine power by pumping the throttle back and forth, but was unable to retain power. Determining that he would not be able to maneuver the airplane to the runway, the pilot opted to perform a forced landing on a dirt path. The airplane landed hard and the right main landing gear collapsed. The pilot reported that there were no mechanical problems with the airplane. Cessna issued guidance to pilots not to perform slips with less than 1/4 tank of fuel; however, the guidance was issued after the airplane's handbook was published. The pilot further stated that if he had maintained a correct airspeed for the approach, it would not have been necessary for him to maneuver the airplane in a slip.