Thank you very much Hans Milton for linking me to that nice demonstration, but I want to tinker with to see if Mathematica can be manipulated so as to find and or solve for the intersection of a RevolutionPlot3D and Graphics3D Polygon.

See this Wolfram Demonstration

Considering yours in the notebook which is `

Manipulate[
 Show[RevolutionPlot3D[-x, {x, -10, 10}, RevolutionAxis -> "Z", 
   Boxed -> False, Axes -> False, Mesh -> None], 
  Graphics3D[{{Opacity[1], 
     White, {Rotate[
       Translate[
        Polygon[{{10, 10, 5}, {-10, 10, 5}, {-10, -10, 5}, {10, -10, 
           5}}], {0, 0, d}], a Degree, {0, 1, 0}]}}}, Boxed -> False],
   ImageSize -> 270], {{d, 0, "Translation"}, -15, 5, 
  Appearance -> "Labeled"}, {{a, 0, "Rotation"}, 0, 145, 
  Appearance -> "Labeled"}]

I tried using DiscretizeGraphics, DiscretizeRegion and RegionIntersection as follows

Manipulate[
 Show[RevolutionPlot3D[-x, {x, -10, 10}, RevolutionAxis -> "Z", 
   Boxed -> False, Axes -> False, Mesh -> None], 
  Graphics3D[{{Opacity[1], 
     White, {Rotate[
       Translate[
        Polygon[{{10, 10, 5}, {-10, 10, 5}, {-10, -10, 5}, {10, -10, 
           5}}], {0, 0, d}], a Degree, {0, 1, 0}]}}}, Boxed -> False],
   DiscretizeRegion@
   RegionIntersection[
    DiscretizeGraphics[
     RevolutionPlot3D[-x, {x, -10, 10}, RevolutionAxis -> "Z", 
      Boxed -> False, Axes -> False, Mesh -> None]], 
    DiscretizeGraphics[
     Graphics3D[{{Opacity[1], 
        White, {Rotate[
          Translate[
           Polygon[{{10, 10, 5}, {-10, 10, 5}, {-10, -10, 
              5}, {10, -10, 5}}], {0, 0, d}], a Degree, {0, 1, 0}]}}},
       Boxed -> False]]], 
  ImageSize -> 270], {{d, 0, "Translation"}, -15, 5, 
  Appearance -> "Labeled"}, {{a, 0, "Rotation"}, 0, 145, 
  Appearance -> "Labeled"}]

but it fails.