Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am interested to know a geometrical structure with maximum surface area and minimum volume. According to me double napped cone may have such property as surface area of a cone is $\pi rl +\pi r^2$ and its volume is $\pi r^2h/3$, where $r$=radius of the base of cone, $l$=lateral height of cone, $h$=height of cone.

share|cite|improve this question
Some fractals suggest themselves – Hagen von Eitzen Apr 19 '13 at 5:20
up vote 1 down vote accepted

I know one of such structures. It is a plane. :-)

share|cite|improve this answer
Yes I m agree with u, but sorry for that I forgot to mention the dimension of the above geometrical structure and actually it should be at least 3-dimensional. – Lakshman Mahto Apr 20 '13 at 10:21
It seems that this restriction does not change the infimum of volume (0) and the supremum of surface area $(\infty)$, but makes them unattainable by bounded convex bodies. But the volume can be done abitrary large, and, simultaneously, the surface area can be done arbitrary small nonzero, by any sufficiently “flat” and “stetched” body. For instance, by the pyramid with the vertices $(-n,-n,0)$, $(-n,n,0)$, $(n,-n,0)$, $(n,-n,0)$ and $(0,0,1/n)$. – Alex Ravsky Apr 20 '13 at 18:09
Thank u for such a beautiful example. As n approaches to infinity volume of the pyramid tends to zero and the surface area approaches to infinity. – Lakshman Mahto Apr 20 '13 at 19:06
Oops, for the volume zero limit the fifth vertex should be $(0,0,1/(n^3))$ instead of $(0,0,1/n)$. – Alex Ravsky Apr 20 '13 at 19:10
Yes actually volume of pyramid is a^2*h/3, if fifth vertex is (0,0,1/n^3) then volume becomes 2/3n. – Lakshman Mahto Apr 21 '13 at 9:27

look up a Menger Sponge on wikipedia

share|cite|improve this answer
Hi, welcome to math.SE. Usually when writing answers here, it's best to add a bit of explanation beyond just "go look here." Even if it's only this much you can also improve it by furnishing the link you are speaking of. You can do this by wrapping the linked text in square brackets and following them immediately with the url wrapped in parentheses. – rschwieb Jun 17 '13 at 13:47
Please try to describe as much here as possible in order to make the answer self-contained. As it stands, this is more of a comment than an answer. – robjohn Jun 17 '13 at 14:34

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.