The shape of the nth in a sequence of random triangle shapes is (x n , y n ). It was shown in [8] and [9] that y n → 0, almost surely, as n → ∞. In this paper we show that x„ converges in distribution, as n→∞, to a random variable x, and we find the probability distribution of x. The convergence in law of x n enables us to complete an argument used in one part of [9].