A random geometric graph, G(n, r), is formed by choosing n points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most r. For a given constant k, we show that (Formula presented.) is a distance threshold function for G(n, r) to have a connected subgraph on k points. Based on that, we show that n− 2/3 is a distance threshold function for G(n, r) to be plane, and n− 5/8 is a distance threshold function for G(n, r) to be planar.