The 6-mile hex is supposedly 32 square miles and the 24-mile hex is 500 square miles. Now, I assume these are approximations, because it doesn’t seem to pan out. I’ve seen discussion on these forums refer to the actual values as various things, but I’d really, really like to know what are the exact values.
I’m actually considering handling realm management outside of the hex context in certain cases, looking directly at square miles instead. However, I’d like to be able to go between the hex and square mile contexts without mysteriously losing square mileage.
A hexagon with a face 3.5 miles on a side has an area of 31.83 square miles (according to Google). (This hexagon actually has 3.49 miles on a side, so its area is 31.64 square miles. In either case, it rounds to 32.)
Scaling up 4x to get a 24-mile hex, we get a face size of 13.96 miles, for an area of 506.32 square miles.
(Note; I am not an Autarch, so there’s always the chance of me being wrong!)
I did a similar thing in my ACKS game and I would advise against using the exact values. Rather, I would suggest using an even broader approximation, one 6-mile hex = 30 square miles, because a 6-mile hex of wilderness requires a stronghold worth 30kgp to secure. By approximating it as a 30-square-mile hex, you get 1 square mile of territory requiring 1 kgp of stronghold in wilderness, 750gp in borderlands, or 500gp in civilized areas. Elegant, no?