You can use the [3D Print Toolbox][1] add-on for that. Activate it in the *Preferences* and it will appear in the Tool Shelf of the 3D View. Top line buttons let you calculate volume and total area.

[![3D Print View][2]][2]


  [1]: https://docs.blender.org/manual/en/latest/addons/mesh/3d_print_toolbox.html
  [2]: https://i.sstatic.net/6O1fQ.png