What's the largest box I can make out of a sheet of wood? I'm planning on building a storage box out of sheet wood, and I'd like to know how to cut up my sheet into 6 pieces to form a box with the largest volume possible. Ideally the box should be flush on the outside, but I imagine the problem is a lot easier to solve when not taking that into account.
In the case of a non-flush box  I'd like a solution that takes the width and height of the sheet and gives me the width height and depth of the interior of the resulting box.
 A: This looks like a perfectly acceptable question. A man has a sheet of material (width W and height H). What is the maximmum volume of a closed box he can make?
We can ignore the thickness of the material initially.
A square box unfolded would be 4 units by 3 units in size - but this would produce a large amount of waste. To reduce the waste, simply divide the sheet into 6 equal sized pieces ( all W/2 x H/3 in size ).
Of the six pieces, four can remain this size but the top and bottom can only be the smallest dimension square. If we assume W < H, then the remaining 2 sides are W/2 square.
Our volume therefore would be, the base area times the height, or
(W/2) * (W/2) * H/3.
If this were a 2.4x1.2 sheet of ply, the maximum volume would be:
(.6*.6) * .8 = 0.48 cubic metres. The waste would be 2 strips of 0.6x0.2
While this produces some waste, we can't distribute that waste evenly to the other 4 sides ( altering one dimension on a box results in four sides changing size). In this woodworking exercise, some of the excess would be used to in the top and bottom because the external dimensions would be 0.6 x (0.6 + 2 x thickness).
A: A 4x8 sheet is 32 sq feet.  For a maximum theoretical volume you would picka cube with equal sides all L^2 in area.  6xL^2=32, or L=SQRT(32/6)=2.3 feet.  The resulting volume is L^3, or 12.32 cubic feet.  Of course, you can't cut the 4x8 sheet into 6 equal 2.3 ft squares, so split the sheet the long way, getting pieces as close to 2.3 ft (in this case 2 ft).  This will allow two ends at 2x2, then 4 equal pieces (tops and sides at 2x3).  The resulting volume is then 2x2x3, or 12 cubic feet, which is as close to the maximum you will get with a realistic cutting pattern.
A: For simplicity (and practical purposes) I will just ignore the saw kerf. Generally speaking, if you are making something out lower grade material it will come in 48 x 96" sheets. Cabinetry type sheets often come in 49 x 97 sheets to account for trim cuts of slightly damaged edges and some saw kerf from your other cuts. Either way, if you use lumber to case the edges as described below, the edges of the box don't necessarily have to meet perfectly.
Rip the 4 x 8 sheet of plywood in half to make two $2 \times 8$ pieces. From both of these strips cut two 3 ft pieces and you are left with one piece 2 x 2 from the end. This will give you 4 pieces @ 3 x 2 for the sides, top and bottom and 2 pieces at 2 x 2 for the ends.
If you want to put stuff in the box, I would probably then use some sort of lumber for all of the outside edge (if you want to put things on the box you can move the lumber to the inside of the box). The width of the lumber depends on what you want to put in the box really, 1 x 2 strapping or 1 x 4 lumber would probably work for the majority of stuff. If you have some scrap 2 x 4 lying around you can also make "feet" for the bottom.
