Here's my hack solution to the fact that RegionIntersection[ <3D mesh Region>, InfinitePlane[...]] does not evaluate. You need to get your cross section into the form of an InfinitePlane, then you find the lines which make up the intersection between the plane and the faces of the mesh region. Problem is now that these lines aren't in any order, so we take all the points making up the lines, and use FindShortestTour to put them in order, and finally make a Polygon out of these.
SeedRandom[42]; region = ConvexHullMesh[ RandomReal[1, {14, 3}] ]; plane = InfinitePlane[{{1., 0.5, 1.}, {0, 0.25, 0}, {2, 0.5, 1}}]; RegionPlot3D[{region}, Axes -> True, AxesLabel -> {"x", "y", "z"}]~Show~Graphics3D[{Red, plane}]
Here we get the polygon,
crossSection = RegionIntersection[plane, #] & /@ MeshPrimitives[region, 2] // DeleteCases[_EmptyRegion] // ReplaceAll[Line :> Sequence] // Flatten[#, 1] & // (#[[Last@FindShortestTour[#]]] &) // Polygon; Graphics3D[crossSection]
Verify that this worked,
Show[ RegionPlot3D[region, PlotStyle -> Opacity[0.5]], Graphics3D[{Red, crossSection}] ]
Then just get the area,
Area@crossSection (* 0.353513 *)
To get more to the exact question in the OP, if you wanted to get the cross section area when you have two points, one being the base, the other being the region centroid, you'd make up the infinite plane thusly
base = (*the origin, why not? *) {0,0,0}; point = RegionCentroid[region]; line = InfiniteLine[{base,point}]; plane = Module[{vec1,vec2,vec3,a,b,c}, vec1=Normalize[base-point]; vec2={a,b,c}/.First@FindInstance[{a,b,c}.vec1==0.&&Norm[{a,b,c}]==1,{a,b,c}]; vec3=Cross[vec1,vec2]; InfinitePlane[point,{vec2,vec3}] ]; crossSection = RegionIntersection[plane, #] & /@ MeshPrimitives[region, 2] // DeleteCases[_EmptyRegion] // ReplaceAll[Line :> Sequence] // Flatten[#, 1] & // (#[[Last@FindShortestTour[#]]] &) // Polygon; Show[ RegionPlot3D[region, PlotStyle -> Opacity[0.5]], Graphics3D[{ Red, EdgeForm[Directive[Thick,Black]], crossSection, Yellow,Sphere[point,.02], Black,AbsoluteThickness[2],line} ],Boxed->False ]
Area@crossSection (* 0.529433 *)
