Denote the Palm measure of a homogeneous Poisson process H
λ with two points 0 and x by P0,x
. We prove that there exists a constant μ ≥ 1 such that P0,x
(D(0, x) / μ||x||2 ∉ (1 − ε, 1 + ε) | 0, x ∈ C
∞) exponentially decreases when ||x||2 tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C
∞ of the random geometric graph G(H
λ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.