"I’ve seen a lot of people go through and calculate the area of a polygon by breaking it up into separate triangles, carefully calculating the area of each, then sum and halve."

I doubt anyone do that. What was described seems a lot more work compared to searching the web for the right formula/algorithm/source-code -- at least in the modern days. A very minimal amount of research will reveal this algorithm mentioned by the author.

That said, it certainly is a very nice algorithm.