# Cylinder Volume Optimization (Calculus)

I have this problem where I have a metal cylinder without the top lid with $V=1\ \mathrm m^3$ and I have to calculate the minimum amount of material to build it, and I do know the way of solving this, but I have a question, I'm basically doing a function of r using the area of the cylinder with the known volume.

My question is, Why do I need to remove the top lid on the surface area? I'm supposed to calculate dimensions, isn't the dimension going to be the same regardless if there is a lid or no lid? Thanks in advance.

• Because removing the lid does change the "surface area" of the cylinder. – Parcly Taxel Dec 8 '17 at 0:48
• if you have a lid, you have a can. if you have not a lid, you have a cup. Two different problems. – cgiovanardi Dec 24 '17 at 14:58

The surface area of a cylinder is simply the sum of the area of all of its two-dimensional faces. removing one of those faces reduces the surface area accordingly.

I think the issue might be with the term "dimensions": while it is true that the radius and height of the cylinder are constant while removing the lid, the surface area is not.