The shape with greatest area to perimeter ratio is a …
The shape with greatest perimeter to area ratio is a …
The shape with greatest total volume to surface area ratio is a …
The shape with greatest surface area to total volume ratio is a …
Fill in the blanks please.
I think these are all set and exact answers, but Idk
Your first and third questions are a circle and sphere. The other two are a little harder to answer.
Let’s say you have a 1x1 rectangle. Area = 4, perimeter = 4. Then consider a 0.5x2 rectangle. Area = 1, perimeter = 5. Half one dimension and double the other again: 0.25 x 4. Area = 1, perimeter = 8.5. You can do this as many times as you like, and the area will stay 1 while the perimeter increases without bound. So there isn’t really an answer to the question. A similar argument applies in 3 dimensions as well.
You eventually end up with an infinite line I suppose. Is that allowed to be an answer? It would have infinite perimeter and zero area.
The Menger Sponge and the Sierpenski Gasket are fractals with zero area/volume and finite perimeter/surface area. Nonetheless, the side length is finite (or at least measurable) so if you theorized one of these shapes with an infinite side length you’d have the answer to 2 and 4. The math gurus here will probably be along soon to tell you how you could increase the cardinality (?) of the infinite measurement that you’re going to get.
I would say the short answer is
Circle
Rectangle
Sphere
Tetrahedron
The longer answer has already been covered, but at some point, an reallllly long rectangle just becomes a line. The point just before that occurs is the shape with the greatest perimeter to area ratio. Same with the tetrahedron.
I think the answer to your fourth one would be Gabriel’s Horn. It has a finite volume and an infinite surface ares, thus it has an infinite surface area to volume ratio.
#2 doesn’t have to be a rectangle. It could be of an shape, or no shape. The only factor is for it to be very long/tall/wide/whatever, and also very short/narrow/thin/whatever. It could be a triangle, an ellipse, a squiggly, anything.
Since when did we start answering homework problems? 
Yesterday, 11:43 PM
LOL
I’m little bit past schoolin age
I’m daydreaming about housing structures.
For a non-fractal, easily graspable answer to the second question, try a hollow star (i.e. from the wingdings font):¶
Increase the number of points to some arbitrarily huge number and then extend them out to some arbitrarily huge distance.
Of course, to answer a somewhat narrower question than you asked, the regular polygon with the greatest perimeter to area ratio is a triangle.
It would really help on a site fighting ignorance to check for accuracy! Typos not allowed. 
A 1x1 rectangle will have an Area = 1 not 4.
You resurrected an ancient thread just to point out a mistake? I have a feeling you will fit in well here. Welcome to the dope.
Sent from my iPhone using Tapatalk
Since we revived this, the classic answer to #2 is a Koch snowflake.
Infinite perimeter, finite area.
NM
[From the irrelevant anecdote desk] Just a note to say this is how we used to calculate land area where I live!
Nowadays there are smart-phone apps which use GPS to calculate area automatically, but when plots of land in the village traded hands two decades ago a low-tech procedure was used:
Approximate the plot of land as a quadrilateral. Measure the sides a, b, c, d. Multiply (a+c) by (b+d) to get the area. In the example this would be (1+1)×(1+1) = 4.
Explanation: the lengths were measured in wa; the resulting area is in square meters. Wa is an ancient unit of length which has since been standardized as 2.0 meters.
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NETA: The “paradox” of Gabriel’s horn (also called Torricelli’s trumpet) attracted great interest from mathematicians and philosophers when it was discovered almost 400 years ago. Its inventor, Evangelista Torricelli, famous for his textbook about possible pitfalls in calculus, spent considerable effort trying to find the (non-existent) flaw in his calculation of its surface area.
OK, I’ll bite: why this formula instead of, you know, some formula that gives the area? (And actually making enough measurements to determine said area.) I am not referring to the factor of 4.
I will grant that that calculation gives an upper bound on the area. But who would pay for more area than they really own?
For a more complicated or expensive lot they might also measure the diagonals, though I don’t know what formula they use in that case.
I live in rural Thailand. 24 years ago when we bought the 8½ acres on which my present house sits, this is how the area was calculated. We walked out the lengths with my father-in-law dropping bamboo(?) rods, each one wa in length. (I think he “palmed” one of the rods but the sellers caught on to that by the next day!) Things are changing quickly, but at that time many had only 3rd-grade education … and the 3rd grades here are not very good.
I could give many examples where performance fell far short of Western standards. But more remarkable are the incidents where these undereducated people display astounding insight or cleverness!
NETA: It wasn’t awfully long ago that the land in my area was mostly government-owned jungle, ravaged by illegal logging. I’ve been told that the default home-stead plot was a circle whose radius was the range of a rifle bullet. However, rather than circles they ended up with nearly rectangular plots, perhaps to facilitate afore-mentioned measurements.