I don’t know your life, but if you’re anything like me you could use a d20 list of dog names for your campaign. I require players to name any animal they acquire, but NPCs frequently have dogs and frequently yell their dogs’ dumb names while the PCs are around.
I needed to figure out how many square miles are in a hexagon. Turns out, nobody wants to give a straight answer when all you know about the hexagon is the distance face-to-face, which is what we all use in the hex and chit world.
After relearning some geometry, this is the simplest possible approximation I can give you: a = (w / 1.732) * (1.5w)
That is, the area is the width “face to face” divided by 1.732, then multiply that by 1.5 times the width.
Example: A Greyhawk 30-mile hex has width 30. It’s area is about 779 square miles.
a = (30 / 1.7320) * (1.5 * 30)
a = (17.3210) * (45)
a = 779.4457
Which means if your stronghold-building wilderness-clearing 9th level Fighter wants a Greyhawk hex all to himself, he’s clearing almost 800 square miles of monsters. Get to it, Gutboy Barrelhouse!
This is the kind of formula I’d put on the second DM screen that you pull out only in extremis. Like how composition books have commonly-used formulae on the inside covers.
Watch this become by far the most useful thing I have ever done 
EDIT: As was pointed out, my formula is off even as an estimate. I checked it against Red Orc’s linked calculator and I was extremely close to being off by doubling the area, so I changed the formula. Regarding Black Vulmea’s formula in the linked Promise City blog post, I guess you have the textbook formula but it looks like the above approximates to yours – close enough for horseshoes anyway. Thank you all for revealing my mathematical decrepitude!
1d30 