I didn’t get all of that done, but I’m working on it now. I still need to make a triangle fan out of each of the four subdivisions, but I suppose I have four partial fans going on, between the center of my world polygon and the edges. I have four daughter polygons going on, but no areas calculated and summed up, yet… I do need at least some subdivision  of these oval pie slices, because as they stand they are two sides of a flat object, not even a low poly polyhedron!

On a side note, my oval-shaped world polygon/ world peach polyhedron is more or less a variation on my square-to-octahedron. In fact, I can turn the faces of any platonic solid (or the polygons of any other polyhedron) into triangle fans with subdivision, but this is what I’m working on/obsessing over right now and it maps fairly easily from 2D to 3D. Latitude is always based on the height of a point inside my world polygon and longitude varies based on height and width; a world rectangle would be better for this, but I hate using that if I can avoid it, because of the polar distortion.

A nice triangle to tetrahedron subdivided to 24-sider is the simplest way of doing all of this (tetra to poor vinnies 24-sided icosahedron equivalent), with the four triangle faces subdivided six ways into a triangle fan.