In this paper we show that every chordal graph with n ver- tices and m edges admits an additive 4-spanner with at most 2n−2 edges and an additive 3-spanner with at most O(n · log n) edges. This signifi- cantly improves results of Peleg and Schaffer from (Graph Spanners, J. Graph Theory, 13(1989), 99-116). Our spanners are additive and easier to construct. An additive 4-spanner can be constructed in linear time while an additive 3-spanner is constructable in O(m · log n) time. Fur- thermore, our method can be extended to graphs with largest induced cycles of length k. Any such graph admits an additive (k + 1)-spanner with at most 2n − 2 edges which is constructable in O(n · k + m) time.