I am not an experienced hex mapper, but your question is interesting to me and I like math. I am always looking up distances and such to strengthen my grasp of the scale I am working with. I am an American though so I generally think in feet and miles. The way I think that works best for me is to use roots of the circumference of the Earth, though of course I guess you could also say that if you have no intention of using a scale as large as a globe you have no need to use this kind of method. Here it is anyway:
C of Earth = 25,000 miles = 131,480,184 feet
(131,480,184 feet)^(1/3) = 509 feet per smallest hex with 3 levels of hexes and a diameter of 3 hexes per every larger hex layer and a hex count of 3 at the highest level (at least in the direction of the circumference or the x axis which is really the same as the equator).
(131,480,184 feet)^(1/4) = 107 feet per smallest hex with 4 levels of hexes and a diameter of 4 hexes per every larger hex layer and a hex count of 4 at the highest level.
(131,480,184 feet)^(1/5) = 42 feet per smallest hex with 5 levels of hexes and a diameter of 5 hexes per every larger hex layer and a hex count of 5 at the highest level.
(131,480,184 feet)^(1/6) = 23 feet per smallest hex with 6 levels of hexes and a diameter of 6 hexes per every larger hex layer and a hex count of 6 at the highest level.
7=14 feet
8=10 feet
9=8 feet
10=6 feet
I'm sure it has to depend on the scale you want to work with. If you have no upper size limit like some planetary circumference or continental width or unsurpassable mountain barrier which you will never travel beyond I suppose it makes sense just to use the simplest and most basic measurement for the smallest hex and decide if you like the look of hexes with 3's, 4's, or 5's for hex diameter. Could be also that I just presented information that isn't new or helpful at all and just got a kick at the expense of your reading time or whatever.
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My math was wrong. Sorry. I should have been doing:
D / (H^H) = F
where D = the maximum distance from one edge of the map to the other, or in the case I presented the circumference of the Earth in feet
H = the number of hexes within a larger hex as well as the number of hex layers
F = the width in feet of a single hex on the bottom layer
So the results are listed below in the form:
H: F
3: 4,869,636 feet
4: 513,594
5: 42,074
6: 2,818
7: 160
8: 8
9: 0.34
Alternatively if you wanted to work from a bottom size up, say F = 100 feet hex diameter (making one hex about the average size of a single family house lot in the United States) you could do:
F * (H^H) = D
to find the maximum size of your map
H: D
3: 2,700 feet (half a mile - distance an out of shape guy like me can run in 5 minutes)
4: 25,600 (5 miles)
5: 312,500 (60 miles)
6: 4,665,600 (884 miles - distance between New York City and St Louis)
7: 82,354,300 (15,600 miles - average distance an American drives in a year)
8: 1,677,721,600 (317,750 miles - almost 13 trips around Earth or about 1.3 times the distance from Earth to the moon)
I've had another thought. Take for example a hex layout where:
H = 5
D = 312,500 feet
F = 100 feet
Perhaps you don't need or want your very top layer at all. After all having just five hexes across your map (about 25 total if you have a squarish mapping area) might not be very helpful. If you want to remove this layer then to give an idea of how many hexes the top layer would contain, it would be H^2. So if you removed the very top layer of this particular map, the map would have a top layer width of 25 hexes, though all other values would be the same.
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