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Algorithm. Appl."],"published-print":{"date-parts":[[2020,2]]},"abstract":"<jats:p> A random geometric graph, [Formula: see text], is formed by choosing [Formula: see text] 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 [Formula: see text]. For a given constant [Formula: see text], we show that [Formula: see text] is a distance threshold function for [Formula: see text] to have a connected subgraph on [Formula: see text] points. Based on this, we show that [Formula: see text] is a distance threshold for [Formula: see text] to be plane, and [Formula: see text] is a distance threshold to be planar. 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