  We give a cluster expansion formula for cluster algebras with principal
coefficients defined from triangulated surfaces in terms of perfect matchings
of angles. Our formula simplifies the cluster expansion formula given by
Musiker-Schiffler-Williams in terms of perfect matchings of snake graphs. A key
point of our proof is to give a bijection between perfect matchings of angles
in some triangulated polygon and perfect matchings of the corresponding snake
graph. Moreover, they also correspond bijectively with perfect matchings of the
corresponding bipartite graph and minimal cuts of the corresponding quiver with
potential.
