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How do I find the size of the root directory? Should I use the root directory entries data field in the BIOS Parameter Block or should I use something else?

Have you checked Microsoft's FAT32 specification?

The root directory is an ordinary directory on FAT32 volumes, so you find its size the same way you find the size of any directory: walk the FAT chain to see how many clusters it uses.

Depends on FAT type. On FAT32, it has it's own cluster link, but for other versions, the size is fixed.