Puzzle:- Maximum volume of liquid held by a unit square folded

note:- folds on to itself are not water proof

Should we consider gravity in only one direction? In which case it's a calculus problem. Is this someone's homework?

real problem is folded paper to filter dirty water, clothes pegs hold corners in place in my latest version

Took me a whole 2nd post to recognize your style, Mr. Burglar!
My offering is the no leak solution. Six faces give the surface area of 6, but for any given surface area, the maximum internal volume is contained in a sphere. So use the formula for a sphere surface area to derive the radius, then use radius to find the volume. Then, simply reform your cubic surfaces into a sphere (he-he).
I am too lazy to work this out - especially while driving in my car.
(Meow)

I should have called it "The Biggest Cup Puzzle" the biggest vessel with an open top (for refilling) made from a unit square but could also have a unit circle version (that could be cut from the square)

Flat earth gravity should be used, where it is all parallel towards an infinite point in another universe

Right angle folding gets a maximum height of 1/6 with a volume of 4/54 ~=~ 0.074074

Hemisphere shaping gets a radius of 1/π with a volume of 2/(3π^2) ~=~ 0.067547

~0.086419 folding at ~0.232 from edge at ~51.17 degrees

~0.086419 folding 0.232180 from edge 0.560008 radians (32 degrees) {2nd try using reduced variables}

Hemisphere can be improved with a Spherical Cap of 139.959 degrees for a volume of 0.0727827, maybe better would be spherical (or parabolic) tray sides/corners