So I am working on redoing my own personal game world and remapping the entire world onto graph paper to represent hexes so if and when anything kingdom related happens in game, I already have it set up for it.
Now, looking to create my legend for the map I scoured the book to find the exact size of a hex so I could extrapolate that number and figure out exactly how big this first continent and subsequent islands and landmarks were.
The book says that a hex is 12 miles from corner to corner, and just under 95 square miles. Great! Right? Wait....
So to figure out the overall area of my map, I decided to figure the side length of a hex, meaning that I divide the square hex into two congruent isosceles right triangles, knowing that the hypotenuse is 12 miles per the book. Then using prior knowledge that I have with isosceles right triangles I know that if I divide that hypotenuse by the square root of 2 (1.414) I can calculate the side length of the hex pretty easily.
This is where things get dumb, I get about 8.486. Now, when I take that 8.486 and square it, I should get just under 95 square miles, right?
Okay, I get 72.021. Wait, what? There's about 23 square miles missing from my kingdoms per hex, multiply by a whole continent and I have a disaster.
Checking my work, I decided to take the sqaure root of 95, and got 9.746, and when I multiply by 1.414 I get 13.781 miles across a hex from corner to corner.
Now my question is: which number is right? The difference comes in about 1.75 miles across the hex. Should I be using the 95 square miles and multiplying that by the number of hexes on my continent to obtain the total area on my world, or should I use the 12 miles and end up with less area overall? Should I use both numbers knowing that the math is wrong because that's RAW? Is this a case where RAW should not be applied in game?
Also, what happens to the nations in Golarion when players start claiming hexes and establishing kingdoms? Do the other nations just let it go, or do they go to war? Is there a lot of unclaimed land I don't know about? How should I handle international policy in my tabletop RPG?