A geodesic triangle Delta(x,y,z) in a metric space is called delta-slim if every point on any side of the triangle lies within delta distance of one of the other two sides. Equivalently, the slimness of the triangle is defined as the infimum of such delta. In this talk, we explore the slimness of geodesic triangles in random graphs, analyzing the expected value of slimness as a measure of how "slim" or "fat" these triangles tend to be. We will also discuss how this analysis provides insights into the geometric and combinatorial properties of random graphs, with connections to broader aspects of metric geometry. Based on joint work with Anna Gilbert.