A dominating set of a graph G is a subset D of vertices such that every vertex not in D is adjacent to at least one vertex in D. A dominating set D is paired if the subgraph induced by its vertices has a perfect matching, and semipaired if every vertex in D is paired with exactly one other vertex in D that is within distance 2 from it. The paired domination number, denoted by γ pr (G), is the minimum cardinality of a paired dominating set of G, and the semipaired domination number, denoted by γ pr2 (G), is the minimum cardinality of a semipaired dominating set of G. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that γ pr (G) ≤ 2 n 4 for any neartriangulation G of order n ≥ 4, and that with some exceptions, γ pr2 (G) ≤ 2n 5 for any near-triangulation G of order n ≥ 5.