We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement of n line segments determined by h endpoints is ⊙(ha(h)). While the previous upper bound, O(nα(n)), is tight for segments with distinct endpoints, it is far from being optimal when n = Ω(h2). Our result shows that the fundamental combinatorial complexity of a face arises not as a result of the number of segments, but rather as a result of the number of endpoints. We generalize our bounds to the case of pseudosegments, and to the case of n chords in a polygon with h holes. Furthermore, our results lead to an improved algorithm for computing a single face in an arrangement of segments that share endpoints.