I don't know what 12 miles from corner to corner mean in the context of the game.
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It seems that 12 mile hexes have been traditional in D&D for decades. The Alexandrian suggests this has to do with how far you can see in normal terrain. It's roughly far away enough that you can't really see details in the next hex, but kinda have an idea what terrain type it might be.
(That site by the way has way more material on hexcrawling than Paizo ever published. But with different opinions about it. Quite interesting for perspective.)
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I used this just now.
https://www.omnicalculator.com/math/hexagon
Long diagonal of 12 miles is 19.3121 km, with 242.24 KM square.
https://www.omnicalculator.com/math/hexagon?c=USD&v=hide:0,d:19.3121!km
Breaking that process down:
The area of any triangle can be found by taking half of the product of its height and its base.
If you choose one of its sides to be its base, then by the Pythagorean theorem the height of any equilateral triangle is √3/2 times the length of that side.
So if we let
S = the length of a side of an equilateral triangle
then the area, a, of that triangle is given by
a = (1/2)·(base)·(height)
a = (1/2)·(S)·[(√3/2)·S]
a = (√3/4)·S²
A regular hexagon whose sides have length S can be made from 6 identical equilateral triangles with sides of length S.
Thus the area of a regular hexagon with sides of length S is given by
A = 6·a
A = 6·[(√3/4)·S²]
A = (3√3/2)·S²
Now, the distance from corner to corner (from one vertex to the vertex opposite from it) of a regular hexagon is twice the length of each side of the hexagon.
So if we let
D = the distance from corner to corner
then we have
D = 2·S or S = D/2.
If we substitute that into the formula above we get
A = (3√3/2)·S²
A = (3√3/2)·(D/2)²
A = (3√3/8)·D²
That is the relationship between the corner to corner length, D, and the area of the hexagon, A.
In this particular case we have D = 12 mi so
A = (3√3/8)·D²
A = (3√3/8)·12² mi²
A = 54√3 mi²
A ≈ 93.53 mi²
Since 1 mi² is approximately 2.59 km² we get
A ≈ (93.53 mi²)·(2.59 km²/mi²)
A ≈ 242.24 km²
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I measured the hexes to be 12 miles across, from one side to another. NOT from corner to corner. There's also a bit of rounding-off of numbers going on as well. Caineach is correct, in any event; a single side of a hex is just under 7 miles long.
The math above is correct but slight correction of side length per James from around 2010 or so, unless hex size has changed.
Area ≈ 127 Sq. Miles or ≈ 330 Sq. Km
Edit: Old Thread
Edit: oops sorry brain went to Kingmaker by default the above is correct mine is only right for the Stolen Lands specifically and you were asking 12 miles corner to corner per the hex rules. Apologies for any confusion caused.
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The PF2 Hexploration rules list a hex size as 12 miles from corner to corner.
But that is also only the recommended or default size. A map could certainly be made with a different hex size.
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The PF2 Hexploration rules list a hex size as 12 miles from corner to corner.
But that is also only the recommended or default size. A map could certainly be made with a different hex size.
Another reason to call the "Stolen Lands" such! PF2e stole nearly 10,000 Sq Miles from them! lol.
Edit: but seriously changed my response above. Didn’t even think about it being made to the size as needed or changes to default hex size and just defaulted to the old Kingmaker value in my brain. Oof.