Keith Lynch wrote: Someone can collaborate right now by giving me some suggestion on how
to figure out the order of the intersections between the plane and the cube edges. Without knowing the order, I don't know which of the pairwise distances to add to get the perimeter (except, of course, in the case of the triangle). The obvious idea is to check for intersections, other than at or beyond the intersections with the cube edges, between two of the lines, but finding intersections between lines in three dimensions is iffy. A miss is as good as a mile.
Here’s one way to do it. The intersection of the 1-skeleton of the cube (i.e. the union of the edges of the cube) with the cutting plane is a set of 3, 4, 5, or 6 points forming the vertices of a convex polygon. If you project those points to the z=0 plane (aka the x,y plane) they still form a convex polygon. By translating the polygon you can get a polygon of the same area and perimeter that contains the origin in its interior. Now we can convert to polar coordinates in the z=0 plane and use theta-values to find out how the points are cyclically ordered. Then computing the perimeter is easy. Areas are even tougher. There's no analog of Heron's formula for
quadrilaterals, pentagons, or hexagons.
I’d suggest using the “shoelace formula” ( https://en.m.wikipedia.org/wiki/Shoelace_formula) to compute the area. Will your talk be viewable on the Internet? It won’t be livestreamed as far as I know, but it will be recorded, and the video will at some point be posted at https://momath.org/math-encounters/ (or maybe https://m.youtube.com/playlist?list=PLBC7544C3215C63A0). Jim