A cross-section of a pyramid

Question

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. This is a repost from this. Can someone help me? Thanks in advance.

Answer 1

The only way an $(n+1)$-gon can be created would be to have a plane slice through all $n$ triangles and the single base. Therefore, the slice cannot be horizontal, since only $n$ sides will be accounted for, nor can it be vertical, since only $(n-1)$ sides can be accounted for or $n=3$ if going through the apex. Therefore, the plane must be diagonal. The following is an example of a diagonal plane through a pentagonal pyramid. Notice how the plane gets a side from each triangle and one side from the base to get a total of 6 sides. However, this doesn't create a regular $(n+1)$-gon since the sides aren't equilateral. In fact, we can say that any diagonal cross section will not be regular since any angle of rotation $\alpha\neq{2x\pi}\ \textrm{rad}$, where $x$ is an integer value, will necessarily skew the lengths of the sides as one end of the plane goes toward the apex and the other end goes toward the base.