Infinite Volume but Finite Surface Area Is there any pathological shape that has a finite surface area but an infinite volume? Sort of like the opposite of a Gabriel's horn.
 A: Aside from my comment about all of $\Bbb R^3$, there is none.  A sphere has the smallest surface for a given volume.  One pathological case would be fractals that have well defined volume but no well defined surface area.  In most cases you would like to say they have infinite area.
A: Nope!
See the "Converse" section of http://en.wikipedia.org/wiki/Gabriel's_Horn .
A: Imagine a sphere outline in an infinite void. If the area within the sphere outline is empty space, and the space outside is solid, it is a 3D shape of infinite volume, and since it continues infinitely, there is no outer edge of the shape to apply surface area to, meaning the surface area is a finite value, on the same spherical plane as the outline.
A: By calculus of variations it is established that a surface area of magnitude $A$ can enclose no more than this maximum volume among all possible shapes in 3-space:
$$ \dfrac{A^{\frac32}}{6 \sqrt{\pi}} $$
when it forms a sphere. So this is a finite limit, cannot go to infinity volume.
For example the 2D Koch curve/snow flakes have infinite perinmeter enclosing a finite area; so also some 3D fractal volumes have infinite surface area can enclose a finite volume. 
