The width and the height are not the same sum
A square is the board overhanging the text block, and it shows on three edges of a finished book: head, tail and fore edge. It does not show at the spine, because the spine edge of the board is set back from the spine by the joint gap so the board has somewhere to swing.
So the height picks up two squares and the width picks up one, less the joint. At the values in the form — a 9 by 6 in text block, an eighth of an inch of square all round, a 0.28 in joint — the board comes out 9.25 in tall and 5.845 in wide. It is half an inch taller than the book and a sixth of an inch narrower. People who expect a board to be bigger than the book in both directions cut it 6.125 wide, and then the fore edge of the case stands a quarter inch proud of the text block and the joint has nowhere to go.
Turn-ins count twice, in both directions
The turn-in is the opposite case. It is on all four edges, so 0.625 in of turn-in adds 1.25 in to the cloth in each direction. The blank at the defaults is 14.946 by 10.5 in, of which 1.25 in each way is cloth that will never be seen from the outside. That is 156.9 square inches of cloth for a book whose two covers are 108 square inches — a third of the blank is turn-in, joint and inlay.
The blank width is worth writing out once, because it is the sum everyone mis-copies: board, joint, inlay, joint, board, and a turn-in on each end. Two boards at 5.845, two joints at 0.28, an inlay of 1.446 and two turn-ins of 0.625 comes to 14.946.
Where the inlay width comes from
The inlay has to bridge the two board edges as well as the spine of the text block, so it is the spine thickness plus two board calipers — 1.25 plus 0.196, which is 1.446 in. Cut it to the spine thickness alone and the case pinches; cut it to the full board-to-board opening of 2.006 in and the joints have no gap left in them at all.
The field for extra inlay is there because a hollow back and a tight back want different things, and because a rounded spine measures differently across the chord than a flat one does. It defaults to zero on purpose. Nothing here knows which structure you are making.
What this cannot tell you
It cannot tell you what square to use, what joint to use, or whether the board you have is right for the book. Those are the choices that make a binding somebody in particular, and they vary by tradition, by the size of the book and by how heavy the text block is. Squares on a small book look enormous at the same measurement that disappears on a folio.
It also has no opinion on materials. Board caliper is whatever your supplier actually sent, which is why the field asks for a caliper reading rather than a product name, and why the answer changes when you measure the board instead of trusting the label.
Questions people ask
Why is the board narrower than the text block?
Because the joint gap is subtracted from the spine edge while only one square is added at the fore edge. At the values in the form the fore-edge square adds 0.125 in and the joint takes 0.28 in away, so a 6 in text block gets a 5.845 in board. If your joint is smaller than your square the board comes out wider instead. The height is a different sum: there the square shows at both ends, so it is added twice with nothing taken off.
How wide should the joint gap be?
This page will not tell you, because it depends on your board caliper, how far the shoulder stands proud after backing, and the tradition you were taught in. The gap has to clear the board edge as the cover swings, so thicker board and a taller shoulder both push it wider. The field defaults to 0.28 in as a placeholder to replace, not as a figure to adopt. Cut one case and open it repeatedly before settling on a number.
How big should the turn-in be?
Also yours. What the page does is take whatever you put in and count it correctly: the turn-in appears on all four edges, so it adds twice to the blank width and twice to the blank height. A 0.625 in turn-in makes the cloth 1.25 in bigger in both directions, and it also decides whether there is any cloth left at the corners once the 45 degree cut is made.
What is the corner cut for?
A square blank turned in at all four edges would double up at the corners into a lump. Cutting the corner off at 45 degrees, set out from the board corner rather than through it, leaves each turn-in a flap that folds against the other. Set the cut too close to the corner and the board shows through; too far out and there is nothing left to fold. The page prints how much cloth the cut leaves on each side so you can see which way you are erring.
Does this work for a rounded spine?
Only if you feed it the right spine measurement. A rounded and backed text block measures less across the spine than the same block does flat, because the arc absorbs the swell, and it is the flat-across chord that the inlay has to match. Work the round out first and use the chord it gives you as the spine thickness on this page.