Calculating a board foot of lumber is just basic math once you know the right measurements to use.
Hardwood lumber is normally figured differently than construction lumber found at lumber yards and home centers. These softwoods used for building material are priced per each or price x length in feet.
Hardwood lumber is priced by the board foot which is the equivalent of 144 square inches or a 1" thick board x 12" wide x 12" long / 144 = 1 board foot. This formula works for any size board, just substitute the actual measurements of the board.
Hardwood lumber usually comes random width and you rip it to the width you want in your shop. The critical measurement when buying hardwood lumber is the thickness. Hardwood thickness measurements are measured in "quarters". A board that can be finish surfaced to 3/4" is considered a 4 quarter thick board. To get a board to clean up at 3/4" usually takes a rough sawn board at least 1" to 1-1/8" thick.
The same as above, it would take a 1-1/4" or more thick board to get a 5/4 or 1" thick board. When measuring for board feet always use the finished thickness size for your calculations. As an example, to find the board feet of a 6/4 or 1/1-2" thick board calculate 1.5 thick x width x length/144 = board feet.
Calculating lumber can also be done quickly with what's commonly known as a log measuring stick or a "Doyle Rule". This stick is commonly used by loggers and lumber buyers to calculate board feet of lumber in standing timber and cut logs when they're delivered to the mill.
For more information about this method of determining a board foot of lumber you'll find a great fact sheet from Ohio State University School of Environment and Natural Resources.