In his paper [2] Lowell J. Hansen found a (nonlinear) recurrence formula for the (sharp constant appearing in the “Ring Lemma” of Rodin and Sullivan [3]. In the following we improve Hansen's result and replace his recurrence relation by a linear recurrence formula leading to a closed formula for the Ring Lemma constant. Moreover, we show that the Ring Lemma constant is a reciprocal of an integer for each n