Permalink Submitted by mathwizard on August 18, 2017

"If you look around the supermarket shelves you will find some cans that have the optimal diameter equals height shape but many do not. Perhaps they just want to look different and stand out on the shelves?"

Well, not really, the real world is more complicated than this. You have to consider manufacturing process: the ends of the cylinders (the two circles) most likely have to be cut from planar material. So one has to consider minimizing wastage as well. We of course know that arranging the circles in a hexagonal array would do the job (this is the "optimal packing"), so it works out that the area to be minimized is really A=4*sqrt(3)*r^2+2V/r. This model would predict that can should be 10% higher than its wide. But in reality we observe differences -- this is true for small cans, like a can of drink. But huge can like a can of paint does tend to have h=2r. A better but more complicated model, taking into account the wielding cost, could explain why its more economical to have huge can be more "squarish", but small can to be tall and thin.

## "If you look around the

"If you look around the supermarket shelves you will find some cans that have the optimal diameter equals height shape but many do not. Perhaps they just want to look different and stand out on the shelves?"

Well, not really, the real world is more complicated than this. You have to consider manufacturing process: the ends of the cylinders (the two circles) most likely have to be cut from planar material. So one has to consider minimizing wastage as well. We of course know that arranging the circles in a hexagonal array would do the job (this is the "optimal packing"), so it works out that the area to be minimized is really A=4*sqrt(3)*r^2+2V/r. This model would predict that can should be 10% higher than its wide. But in reality we observe differences -- this is true for small cans, like a can of drink. But huge can like a can of paint does tend to have h=2r. A better but more complicated model, taking into account the wielding cost, could explain why its more economical to have huge can be more "squarish", but small can to be tall and thin.