We give asymptotically exact values for the treewidth tw(G) of a random geometric graph G(n, r) in [0,\sqrt{n}]^2. More precisely, we show that there exists some c1 > 0, such that for any constant 0 < r < c1, tw(G) = \Theta(log n/log log n), and also, there exists some c2 > c1, such that for any r = r(n) \geq c2, tw(G) = \Theta(r \sqrt{n}). Our proofs show that for the corresponding values of r the same asymptotic bounds also hold for the pathwidth and treedepth of a random geometric graph.