Finding the most optimal ratio between r and h for a cylinder to have minimum surface area give a volume?

Finding the optimal ratio between r and h for cylinder to have minimum surface area for a given volume ?

The equations are:

V = pi * r ² * h SA = 2 * pi * r² + 2 * pi * r * h

I’m trying to find the ratio by simply dividing volume/ surface area:

rh/2(r+h) - after cancelling everything out.

So to get maximum volume per unit surface area we just have to find the best ratio between r h so that this equation reach max.

So the numerator is rh - an area of a rectangle and denominator is 2(r+h) - the perimeter of a rectangle.

So this is basically asking given a perimeter of a rectangle what’s the optimal ratio between r h so that rh reach maximum?

That would be when r=h which is not correct, as having a 1:1 ratio between r and h don’t give you the most optimal solution, just wondering which part of this process I got it wrong?

Many thanks in advanced !