@anorton The explanation of the asymmetry you give now (in terms of the triangle formed by the segment with the origin) corresponds to the area formula from wikipedia. However, a single term is your area formula gives the area of a trapezoid, which is a bit more asymmetric, but much more useful in practice.

I say it is more useful in practice, because something like this can be used to compute local densities of a polygonal pattern on a uniform 2d grid. Such polygonal patterns occur in semiconductor layouts during data preparation for manufacturing of the masks (and other pattern transfer steps). Here, you have to do additional clipping operations with these trapezoids, and clipping these trapezoids is simpler than clipping triangles.

Just to point out, Stack Edit has markdown and mathjax support so it's a good option to make drafts of the blog posts