```
''' cylinder_surface_min.py
calculate the height to radius ratio of a minimum surface cylindrical
container (can)
r = radius
circle_area = pi * r**2
circle_circumference = 2 * pi * r
h = height
cylinder_volume = pi * r**2 * h
surface area of a closed cylinder:
the area of the top (pi * r**2) +
the area of the bottom (pi * r**2) +
the area of the side (2 * pi * r * h)
or
cylinder_surface = 2 * (pi * r**2) + (2 * pi * r * h)
let the cylinder_volume be a constant of 500 ml (cubiccentimeter)
cylinder_volume = 500
using the cylinder_volume formula express h in terms of r ...
h = 500/(pi * r**2)
now substitude h in the cylinder_surface formula
(this expresses the cylinder surface in terms of r)
cylinder_surface = 2 * (pi * r**2) + (2 * pi * r * 500/(pi * r**2))
tested with Python34 by vegaseat 5apr2015
'''
from math import pi
import pprint
# rough calculation of the radius range to achieve a minimum surface
# create a list of (surface area, radius) tuples
mylist = []
for r in range(1, 11):
cylinder_surface = 2 * (pi * r**2) + (2 * pi * r * 500/(pi * r**2))
mylist.append((cylinder_surface, r))
#pprint.pprint(mylist) # test
''' we see that the surface minimizes between r=3 and r=5
[(1006.2831853071795, 1),
(525.1327412287183, 2),
(389.88200109794957, 3),
(350.5309649148734, 4),
(357.0796326794897, 5),
(392.86133772513176, 6),
(450.7332229089426, 7),
(527.1238596594935, 8),
(620.0491209926575, 9),
(728.3185307179587, 10)]
'''
# since range() takes only integers let's use a while loop
# then we can increment in smaller steps
r = 3
surface_list = []
while True:
cylinder_surface = 2 * (pi * r**2) + (2 * pi * r * 500/(pi * r**2))
surface_list.append((cylinder_surface, r))
r += 0.1
if r > 5:
break
#print(min(surface_list)) # test
''' the minimum surface of a 500ml can is reached at r = 4.3
(348.7342358646343, 4.3)
'''
min_surface = min(surface_list)[0]
sf = "Minimum surface of a 500ml can = {:0.1f} square centimeters"
print(sf.format(min_surface))
radius = min(surface_list)[1]
print("Radius of 500ml can = {:0.1f} centimeters".format(radius))
height = 500/(pi * radius**2)
print("Height of 500ml can = {:0.1f} centimeters".format(height))
# true for any volume ...
sf = "Ratio of height to radius of a minimized surface can = {:0.1f}"
print(sf.format(height/radius))
''' result ...
Minimum surface of a 500ml can = 348.7 square centimeters
Radius of 500ml can = 4.3 centimeters
Height of 500ml can = 8.6 centimeters
Ratio of height to radius of a minimized surface can = 2.0
'''
```

The article starter has earned a lot of community kudos, and such articles offer a bounty for quality replies.