  We study the graph structure of large random dissections of polygons sampled
according to Boltzmann weights, which encompasses the case of uniform
dissections or uniform $p$-angulations. As their number of vertices $n$ goes to
infinity, we show that these random graphs, rescaled by $n^{-1/2}$, converge in
the Gromov--Hausdorff sense towards a multiple of Aldous' Brownian tree when
the weights decrease sufficiently fast. The scaling constant depends on the
Boltzmann weights in a rather amusing and intriguing way, and is computed by
making use of a Markov chain which compares the length of geodesics in
dissections with the length of geodesics in their dual trees.
