Optimum fitting for flanges in a rectangular plate I have a $2500~\text{mm}\times6300~\text{mm}\times25~\text{mm}$ (width $\times$ length $\times$ thickness) steel plate I want to cut flanges of diameter $235~\text{mm}$ can anyone please suggest
$1)$ How many flanges would fit in this plate?
$2)$ A method of cutting circular flanges so that wastage is minimum?
$3)$ A generalized algorithm that would help me calculate this for any plate size?
(P.S: I have heard about the packing problem but i am unable to understand it)
 A: If you arrange the flanges in a rectangular grid, you'll get 26 rows and 9 columns. This is a total of 234 flanges. If you arrange the flanges in a staggered fashion (aka "hexagonal close packed"), you'll get 26 rows and 12 columns or a total of 312 flanges.
There is no generalized algorithm known (to me) which can do much better. Packing researchers seem to be focused on packing circles in squares rather than in rectangles.
There is an online circle packing calculator available for hexagonal packings. Note that you have to scale dimensions by 235 as the calculator assumes unit diameters.

You might have to add a safety margin to the 235mm to take into account the cutting width.
A: When you can get a lot of flanges out of the rectangle, the optimal packing is hexagonal.  You can see it .  If you put one row along the long edge of your plate, you will get $\lfloor \frac {6300}{235}\rfloor = 26$ along the bottom row.  You get $\lfloor \frac {2(2500-235)}{235\sqrt 3} +1\rfloor=12$ rows.  Each row has $26$, so you get $312$ of them.  If you put the straight edge along the short direction, you get a first row of $10$ and each row has $29$ for a total of $290$.  The first is better.  When the flanges are larger compared to the sheet, it gets more complicated.  With other plate dimensions, the intervening rows might be one flange short.
