penPts = {Cos[#], Sin[#]} & /@ Range[0, 2 Pi, 2 Pi/5][[1 ;; -2]];

 tau = (2 Sqrt[5])/(5 + Sqrt[5]);

 Graphics[{Blue, Polygon[penPts], Red, PointSize [0.03],
 Point[penPts[[2]]*tau + penPts[[5]]*(1 - tau)], Green,
 Line[penPts[[{1, 3}]]], Line[penPts[[{2, 5}]]]}]

pentagram[pts_] :=
  Riffle[pts, #] &@(pts[[# + 1]]*tau + (1 - tau)*
        pts[[1 + Mod[# + 2, 5]]] & /@ Range[0, 4, 1]);


Graphics[{Red, PointSize [0.03], Point[pentagram[penPts]], Green,
  Opacity[0.5], Polygon[pentagram[penPts]]}]

ind = PolyhedronData["Dodecahedron", "FaceIndices"];
vert = PolyhedronData["Dodecahedron", "VertexCoordinates"];
Graphics3D[ Polygon /@ pentagram /@ (vert[[#]] & /@ ind)]

Graphics3D[Polygon /@ Reverse@*pentagram /@ (vert[[#]] & /@ ind)]

MeshCells[
 DiscretizeGraphics@ Graphics3D@ Polygon@ pentagram@ vert[[First[ind]]], 2]
(*
  {Polygon[{1, 9, 10}], Polygon[{3, 1, 2}], Polygon[{4, 1, 3}], Polygon[{5, 1, 4}],
   Polygon[{6, 1, 5}], Polygon[{7, 1, 6}], Polygon[{8, 1, 7}], Polygon[{9, 1, 8}]}
*)

Graphics3D[{Polygon[#],
    MapIndexed[Text[Style[#2, "Label", Bold, 16], #1] &, #]} &@
  RotateLeft@pentagram@vert[[First[ind]]]
 ]

DiscretizeGraphics@ Graphics3D@ Polygon@ pentagram@ vert[[First[ind]]]
DiscretizeGraphics@ Graphics3D@ Polygon@ Reverse@ pentagram@ vert[[First[ind]]]

Graphics3D[{Polygon[#],
    MapIndexed[Text[Style[#2, "Label", Bold, 16], #1] &, #]} &@
  RotateLeft@ pentagram@ vert[[First[ind]]]
 ]