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This question isn't related to my math classes since I'm not taking any geometry, but I came up with the question of whether there's any solid where $V/S$ is a constant. If there was one, you could shrink or expand the solid and the ratio wouldn't change.

The only progress I've made is deciding that the solid can't be a prism. If $A$ is the base area, $P$ is the perimeter of the base, and $h$ is the height of the prism, then $$V=Ah$$ $$S=2A+Ph$$
Dividing $V$ by $S$ and solving for $h$ gives you $$\frac{V}{S}=\frac{Ah}{2A+Ph}=c$$ $$Ah=c(2A+Ph)$$ $$Ah-Pch=2Ac$$ $$h=\frac{2Ac}{A-Pc}$$ So if I have a prism and change its height without changing the base, the ratio $c$ would have to change. I'm guessing such a solid doesn't exist, because I feel like the two formulas always involve different powers (like $s^2$ and $s^3$ for a square), but I'm wondering if there's a way to prove it's impossible or find a solid that does satisfy the condition.

Edit: A comment pointed out that it might make more sense to assume the shape only changes proportionally, so every dimension increases by the same factor. Answers that use this assumption would also be appreciated.

MumboJumbo
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    Almost certainly of interest: https://math.stackexchange.com/questions/625/why-is-the-derivative-of-a-circles-area-its-perimeter-and-similarly-for-sphere – Andrew D. Hwang Nov 17 '22 at 01:54
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    Often when we talk of "expanding" a solid we really mean that we replace the original solid with one that has all the same proportions but is larger in every direction. So we wouldn't change the height of a prism while leaving the base unchanged; we would expand both by the same percentage. If you're willing to change the proportions of the object when you "expand" it, would you be willing to make the base slightly smaller while increasing the height? You might be able to get a constant $V/S$ ratio that way. Whether that counts as "a solid" is a question, however. – David K Nov 17 '22 at 04:17
  • This is a great question. The answer is "none". For conventional objects, $V/S \sim L$, where $L$ is some characteristic length scale of the object. However, there are many pathalogical objects which do not share this behavior. If you want to explore pathologies, a fun start is fractal objects: https://en.wikipedia.org/wiki/Fractal – kevinkayaks Nov 17 '22 at 21:29

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One way to see this will not work with anything we would consider a normal solid is to note that $V$ has dimensions of length$^3$ while $S$ has dimensions of length$^2$. The ratio therefore had a dimension of length so it depends on the unit we measure in. If you double the size of a solid the volume goes up by a factor $8$ and the area a factor $4$, changing the $\frac VS$ ratio by a factor $2$.

In pathological solids you might consider Gabriel's horn, which has finite volume and infinite surface area. No matter what the volume, you can say $\frac VS=0$ but usually we do not consider infinity a number so you can't divide by it.

Ross Millikan
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