So… a player in my current Pathfinder 1E game made an observation that made me scratch my head a bit. He came to the conclusion that the item Minor Bag of Holding is “smaller on the inside”.
To be honest I had never paid this item, originally printed in Pathfinder Chronicles: Classic Treasures Revisited, any attention. It has a purchase cost is 1,000 gp and a meager 50 pounds or 6 cubic feet of capacity. At least to me a Minor Bag of Holding never seems to not be a very worthwhile investment, when I could just wait a while and buy a Handy Haversack or a Bag of Holding Type I. Due to this myopia on my part, I never noticed something interesting about this item and it breaks down as follows:
- Like a normal bag of holding it “appears to be a common cloth sack”.
- Cloth sacks are basically cylindrical in shape while full.(not really, but more on this later)
- “It measures 2 feet by 4 feet”
- The player made the assumption that the above dimensions were for an inflated/opened bag of holding rather than a flat one. I also made this assumption at first.
- The volume of a cylinder is given by this formula: V=πr2h
The conclusion that the player and I reached based on the above above was that the volume of a bag with a Minor Bag of Holding’s listed dimensions should be 12.57 cubic feet, better than double the volume of the internal extradimensional space listed for the items. From this perspective the item’s only really use would be as a weight saving measure.
Not quite satisfied with this answer I decided to dig a bit deeper. Remember, that I noted that a filled sack is not really a cylinder and that the above is based on the assumption that the measurements are for an inflated bag of holding. I began to wonder if perhaps the given dimensions were for a flat bag. This of course made me scratch my head and wonder how to calculate such a thing, but by the might of Google I found this: The Paper bag problem . In short the paper bag problem is the question of how to calculate the maximum filled volume of a rectangular bag, and the answer for a bag with an open edge is the following: V=w3(h/(πw)-0.071(1-10(-2h/w)))
Plugging everything in we come up with: 23(4/(π2)-0.071(1-10(-8/2)))= 4.53 cubic feet
We end up with 6 cubic feet of space residing inside of a 4.53 cubic feet bag. It turns out that the Minor Bag of Holding is indeed bigger on the inside than its external dimensions would indicate.
The moral of this story is that things that seem quite simple can be complicated once you start digging into them.