Which solids have least volume to surface area ratio?

Question

Sphere has the highest volume to surface area ratio. Which solid shapes have the least volume to surface area?

By solids I meant regular 3-D geometrical shapes like cube etc..

Answer 1

An extremely thin plane, like a piece of paper maybe?

For example if you had an object that measured 10'x10'x0.01' its total volume would be 1 cubic foot, but its surface area would be
2(10x10) + 4(10x0.01) = 200.4 square feet.

That would give you a ratio of 200.4 feet.

0.01 feet is pretty thick. If we made it 1% as thick, that would give us a ratio of 20,000.004 feet. You see where I'm going with this.

Answer 2

That would be the heaviest solids if weight and shape were the identity characteristics. Take 2 objects of identical weight and shape. The densest one, such as plutonium would have least volume. Quite recently
2 new elements have been discovered at Berkeley -
116 and 118 (identified by their number of protons).
But if you are looking for a shapewise answer then it
would be a flat sheet. The flatter, the better to fulfill your
requirement to minimize volume to surface. And I claim
it would not matter what 2 dimensional shape it took so long as it was flat. As a thought experiment, imagine that
an object could be flattened down to 1 molecule thick.
You would have all molecules exposed and the area would
be the area of 1 molecule times the number of molecules
in the object. So the 2 dimensional shape would not matter. Flatness is the key.

Answer 3

I am not quite sure about which particular regular geometrical shape will it be...but definitely a hollow cube is going to have a much larger Surface are(S.A) than a closed one...similarly a hollow cone can have much larger S.A than an enclosed one ...or for that matter even a cuboid...there are so many regular geometrical petterns you can think of...regular tetrahedron,octagedron,hexagonal....and so on...but to find out which has a least S.A would require a comparative study of all the figures having same volume,express their volumes in terms of area and then equate their derivatives to zero and follow the method of maxima/minima...to determine the least of all...firstly this method is too cumbersome...and next...there is no end to the set of regular geometrical patterns that a sold might have....mathematicians and scientists are still cracking their brains over the search of more regular geometric structures..that a solid might assume.....

Answer 4

Well now lets see what the question really is lol? If you are talking surface area like a 6" dia sphere, a 6"x 6" x 6" cube, or a 6" x 6"x 6" equal lateral triangle. The the Triangle would be the least surface area.

Answer 5

Maximize surface area by increasing the amount of lumps!

Star shaped solids have more surface area to volume.

If you developed a solid with a fractal surface, you could have infinite surface area in a finite volume! Look at the geometric shapes in the mathworld wolfram site and see how the circumference can increase but the enclosed area would reach an outer limit. Apply this 3 dimensionally!

And I like K Biz's answer as well...

two dimensional (or as close as possible to it) object will have very high surface area to volume

but I like my infinite surface area better :)

Answer 6

According to the chaos theorem, the fractals shows that it is possible to get a solid with an infinitely large surface area and finite volume. this is not a definite solid it is a general case. According to this fact the requires ratio will be a zero

Answer 7

mmm. as of yet, mathematically impossible question. You can imagine a plane as having that, or as others suggest, some 3D object with highly folded ridges along the surface...

However, it is impossible to define because you can always have a thinner rectangular plane stretching out farther and farther to infinity, or add one more ridge, etc.

However, if you find an equation that will give us a conclusion other than infiniti, I'm pretty sure you'll get a Nobel Prize in Mathematics for that. :)

So yah, the answer is that it does not exist (or infinit, whatever).

Answer 8

I think it'd be solid shapes with numberous hollow cut throughout the object. One example of a light object which gives you maximized surface area is the lung.

Answer 9

volume could only be determine if dimensions are given from any object of your choice. Only then you could answer your question.
Sample: a ring box 1/2" x 1/2" x 1/2 will have less volume than the Louisiana Superdome.

Answer 10

This is the type of question you might have in a differential geometry class. However, the way you stated it makes me think that there is no solution. For a typical minimal surface problem you would have some sort of boundary condition.