Given $F \subset [0, 1]^2$ and finite, let $\sigma(F)$ denote the
length of the minimal Steiner triangulation of points in F. By showing
that minimal Steiner triangulations fit into the theory of subadditive and
superadditive Euclidean functionals, we prove under a mild regularity condition
that $$\lim_{n \to \infty} \sigma(X_1,\dots, X_n)/n^{1/2} = \beta \int_{[0,
1]^2}f(x)^{1/2} dx \c.c.,$$ where $X_1,\dots, X_n$ are i.i.d. random variables
with values in $[0, 1]^2$, $\beta$ is a positive constant, f is the
density of the absolutely continuous part of the law of $X_1$ , and c.c.
denotes complete convergence. This extends the work of Steele. The result
extends naturally to dimension three and describes the asymptotics for the
probabilistic Plateau functional, thus making progress on a question of
Beardwood, Halton and Hammersley. Rates of convergence are also found.