A tangentially cut rod packing ring is provided. The tangentially cut rod packing ring comprises a first ring formed from a plurality of segments. Each of the segments has a portion of two interfaces where each interface slidably engages an interface of an adjacent segment. One interface terminates at a leading surface. One interface has a stop surface. The leading surface is originally separated from the stop surface by a gap. As the interfaces slidably move, the gap lessens until the leading surface abuts the stop surface. The tangentially cut rod packing ring also comprises a second ring formed from a plurality of segments. The second ring has a first portion and a second portion. The first portion forms as shelf on which the first ring sits. The second portion surrounds the first ring. An elastic member in a groove on the outer surface of the second portion provides a compressive force on both the second ring and the first ring.