How to decrease surface area of a box and increase volume?

Question

Does anyone know of a strategy to decrease the surface area of a regular shoebox but increase the volume. We can unfold and reconstruct the box im just not sure what to do to decrease the surface area without the volume getting smaller too...

Answer 1

A simpler strategy is to rebuild the box to be closer to a cube (all sides equal). A cube has the smallest surface area of rectangular solids.

The best of all it to remake it as a sphere, that would be as low as you can get.

Answer 2

there are a couple of links below that can help with the math

you can either use calculus, or plot the parameters

the basic technique is to consider a flat sheet of length L and Width W, with squares cutout of each corner of side x

the shoebox that is formed has

volume V(x) = (L-2x) * (W-2x) *x

and Area A(x) = L*W-4*x^2

example below

L W x Area Volume

20 10 0.5 199 85.5

20 10 1 196 144

20 10 1.5 191 178.5

20 10 2 184 192

20 10 2.5 175 187.5

20 10 3 164 168

20 10 3.5 151 136.5

20 10 4 136 96

20 10 4.5 119 49.5

20 10 5 100 0

as you can see there is an area where the volume is increasing where the area is decreasing