  The Brownian triangulation is a random compact subset of the unit disk
introduced by Aldous. For $\epsilon>0$, let $N(\epsilon)$ be the number of
triangles whose sizes (measured in different ways) are greater than $\epsilon$
in the Brownian triangulation. We determine the asymptotic behaviour of
$N(\epsilon)$ as $\epsilon \to 0$.
  To obtain this result, a novel concept of "large" dislocations in
fragmentations has been proposed. We develop an approach to study the number of
large dislocations which is widely applicable to general self-similar
fragmentation processes. This technique enables us to study $N(\epsilon)$
because of a bijection between the triangles in the Brownian triangulation and
the dislocations of a certain self-similar fragmentation process.
  Our method also provides a new way to obtain the law of the length of the
longest chord in the Brownian triangulation. We further extend our results to
the more general class of geodesic stable laminations introduced by
Kortchemski.
