Clear[volumetricPlot]

volumetricPlot[latticeType_]:=Module[{img=LatticeData[latticeType,"Image"],r=LatticeData[latticeType,"PackingRadius"]},Show[img/.Sphere[pt_,r_]:>{},Map[RegionPlot3D[(EuclideanDistance[{x,y,z},#]<r),{x,-1,1},{y,-1,1},{z,-1,1},Mesh->False,PlotStyle->Opacity[.5]]&,Cases[img,Sphere[pos_,_]:>pos,Infinity]]]]

volumetricPlot["FaceCenteredCubic"]
volumetricPlot["HexagonalClosePacking"]