A rectangle-of-influence drawing of a plane graph G is a straight-line planar drawing of G such that there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of any edge. In this paper, we show that any given 5-connected plane graph G with five or more vertices on the outer face has a rectangle-of-influence drawing in an integer grid such that W + H ≤ n - 2, where n is the number of vertices in G, W is the width and H is the height of the grid.