A straight-line grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straight-line segment without edge crossings. It is well known that a planar graph of n vertices admits a straight-line grid drawing on a grid of area O(n2). A lower bound of Omega(n2) on the area-requirement for straight-line grid drawings of certain planar graphs is also known. In this paper, we introduce a fairly large class of planar graphs which admits a straight-line grid drawing on a grid of area O(n). We also give a linear-time algorithm to find such a drawing. Our new class of planar graphs, which we call "doughnut graphs," is a subclass of 5-connected planar graphs. We also show several interesting properties of "doughnut graphs" in this paper