For solvent accessible surface area calculation, use the DS Client menu command: Structure > Solvent Accessibility…
The results are reported in the molecule data table: in Molecule tab for the total SAS, in AminoAcid tab for the per residue SAS.
The Sidechain Solvent Accessibility is also reported in the AminoAcid tab.
Note: If you select the Hydrophobic group (in Hierarchy view, under Protein Groups) first, then do the calculation, the reported SAS area will be for the hydrophobic residues only.

More details are in the DS Help topic: Working with Discovery Studio > Working with structures and surfaces > Tasks - Structures > Calculate solvent accessibility
Note: The solvent accessible surface area calculation is a DS internal method, there’s no publication but the Theory section in DS Help gives a very detailed description of the methodology. See the page:
Working with Discovery Studio > Working with menu tools > Structure menu > Structure | Solvent Accessibility... > Solvent Accessible Surface