  The Ruppeiner geometry and the fluctuation for a four-dimensional charged AdS
black hole in the framework of the extensive thermodynamics have been
investigated. By fixing the AdS radius of Vissers construction, the extensive
thermodynamics of the RN-AdS black hole is established, and the central charge
plays the role of particle number. With the thermodynamic variables density, we
reexamine the phase transition of the black hole, and the result will hold for
the black holes under different gravitational models. Moreover, we have studied
the Ruppeiner geometry and the black hole fluctuation, and show that the
increase of central charge will reduce the fluctuation probability.
