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Dale is an engineer, so he use the "correct" way to calculate a hexagon size in his guide, i.e. he use the side.
Who wrote the Paizo rules is a gamer, and use the standard system used in all boardgames, calculating the distance from a side to the opposite side of the hex (or if you prefer, from the centerpoint of one hexagon to the centerpoint of the adjacent hexagon), so the disparity.
You can find some comment about that in this section of the forum and in the thread speaking about Book of the River Nations.
As Urath DM said, there was considerable confusion over what "12 miles across" meant. The first version of the our rules (Exploration and Kingdom Building) had the measurement being 12 miles from side to side. As you pointed out Laithoron, that comes out to 125 mi^2, not 150. The only way I was able to get ~150 miles^2 with a 12 miles across statement is to assume a square instead of a hex. That would give 144 mi^2.
We looked for clarification on the boards, but ultimately we needed to make a decision and we chose to go with 12 mile sides. At the time we felt that was what Paizo had originally intended. That comes out to ~375 mi^2. That is the official ruling for this book.
If you prefer to adjust the number for your own game, it has little bearing on the way the kingdom building aspect is played. In the same way that the kingdom's population is simply a fluff number, the dimensions of a hex are more fluff then of game altering mechanics. For example the travel time across a hex is a static number, despite the distance across a hex varies widely depending on how you are measuring it. A 12 mile side is 24 miles when measured from corner to corner, yet the travel time remains the same. So if you choose to call the hex size different but leave the travel time the same, it has little impact on game play.
Sorry for the confusion.
The Book of the River Nations doubles the width of hexes (and quadruples the area of hexes).
The Kingmaker book said 12 miles from corner to corner (or 6 miles a side). James Jacobs once clarified this as 12 miles center-to-center (Approximately 7 miles a side). Both of these are much less than The Book of the River Nations' statement of 12 miles to a side.
This causes conflict with the Pathfinder overland travel rules (basically doubles the speed of travel), whereas the Kingmaker version is a mere simplification of the overland travel rules.
The Book of the River Nations also misrepresents the size of The Stolen Lands as larger than Washington State, whereas its supposed to be about the size of Maine. It also makes city districts seem like they take up much less space in a hex than they really do.
In other words, this small mistake "breaks" a lot of the assumptions that go into the Kingmaker AP.
Any chance of fixing this in the PDF?
This was asked upthread I said then that I was not going to. There are several reasons why. I don't know if I shared them all in one place before so I'll share them all here:
1) It doesn't change any of the actual game mechanics itself. The travel time table is identical to the one in the Adventure. If you want to change that for your home game it makes no difference to the game itself. None. If you want to change it for your own game, rest assured that your game experience will not change. Just like the actual population number, no mechanics are actually dependent upon the size of the hex or the area of the kingdom. Simply the number of hexes your kingdom occupies.
2) This is not a slam on Paizo or anyone that works there but I do not believe that anyone there is either a mathematician or civil engineer. Infact their open calls for game designers/developers typically calls for an english or a history degree. (I'm an engineer by day job and by degree.) The way you calculate area of a hexagon is to measure a side. So my natural assumption was that the 12 miles corner to corner was that it was a typo and they actually meant 12 miles per side. Jacobs clarification came after (or atleast I didn't hear about it until after) the book had already gone to print (and mathematically speaking, you don't measure the distance between points A and B to get something that B has nothing to do with, it is poor mathematics). Before it went to print, however, I ran through ALOT of calculation models before publishing the book (including models with 6 mile sides and 7 mile sides, because I did question exactly what they meant) and I found that the 12 mile sided hex worth of farms was closer to providing the consumption reduction (the actual number was like 11.6 something miles, IIRC). Mind you, the calculations assumed alot of things that simply aren't true like the growth rate was constant from year to year and all farmers were equally skilled, but it is as historically accurate as I can make it. That was the same reason why I switched the Warden and the Marshall, historical accuracy.
3) The Book of the River Nations has sold over 1000 copies. There are only 2 other Pathfinder Compatible books that I know of that have also passed that threshold (Tome of Horrors Complete and Psionics Unleashed).
There are simply to many copies of the book out there for me to invalidate for something that neither is historically accurate nor changes the way the game is played. If there ever is a second edition of the book, we are on the list of things to look at, but that will depend on if any future books use the kingdom building rules (*cough*Shadowsfall*cough) and their relative popularity and so forth. All I can say that we'll see. But for this version, the way it is is the way it is.
This should resolve the question for a couple of months, till the next batch of gamers start Kingmaker. :D
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Standard rules: here
1 day of walking, with a base speed of 15 = 12 miles, base speed of 20 = 16 miles, base speed of 30 = 24 miles, base speed of 40 = 32 miles
Then you apply the terrain modifier, off road, plains x3/4, hills x1/2, forest x1/2.
Kingmaker exploration rules:
Traveling (Time to cross 1 hex)
Party Speed Plains All Other Terrains
15 feet 11 hours 16 hours
20 feet 8 hours 12 hours
30 feet 5 hours 8 hours
40 feet 4 hours 6 hours
50 feet 3 hours 5 hours
Exploring (Time to fully explore 1 hex)
Party Speed Plains (Forest or Hill) (Mountain or Swamp)
15 feet 3 days 4 days 5 days
20 feet 2 days 3 days 4 days
30 feet 1 day 2 days 3 days
40 feet 1 day 1 day 2 days
50 feet 1 day 1 day 1 day
(sorry, I never remember what is the command to format it as a table)
I think that the exploration time is a bit on the generous side, but it can be useful to justify why some stuff that you haven't noticed on the first exploration can be found later.