There was a great post a while back on the recommended size of the board based on piece size. I think Goldendog wrote it. I think for a 4" King you want either 2 124 or 2 1/2" squares.
Need help deciding chessboard

I know about the 'optimal' size being between 2 and 2,5 inch. I would love to see some visuals on that.

There are some pictures in this thread: http://www.chess.com/forum/view/chess-equipment/best-board-size
Personally, I think 2 inches is too small for a 1.75 inch king base. I would prefer 2.25 or 2.375. Your option of 2.4 inches would be a good choice.
Hi all,
I would like some help in deciding on the size of my chessboard. I am getting one custom made. I have some samples in on the wood types, see here my choice: http://tinyurl.com/lvvp7j
My king height is 4 inch, with a diameter of 1 3/4 inch.
However, I don't know yet the field size i'd like to have. I now have a board with field size 2 inch. How ever I'm thinking that maybe a fieldsize of 2,4 inch might be nicer.
Having visual feedback would be absolutely great. If you could all help me by showing some pictures of chessboards/sets that resamble the above...
Hope to here (and see) many reactions. Many thanks in advance!