This is based on the question about maximizing the volume of an open box being formed from a square with corners cut.
Original question is here Optimisation question
edit: added above picture for extra clarity about the case of a square base.
I also solved this for the base being a triangle, pentagon and hexagon, the trig gets a little hectic. Spoiler alert - the answer in all cases ended up with $x$ being $1/6th$ the length of the original shape's side.
My question is how to prove if is this the case for all regular polygons.
In this low quality sketch, the cuts are along the green lines, making the rectangular flaps that fold up to make the triangular based box. Similar kite shapes need to be cut for the other shapes.