Spheres maximize enclosed volume I've heard many times, that among all possible $2D$ shapes with given perimeter, circles enclose the largest area and among all $3D$ shapes with a given surface area spheres enclose the largest volume.
Why is that?
Does the pattern extend to higher dimensions?
 A: Here's just a little bit of intuition to understand why circles enclose the largest area.
Assume that a given perimeter = $P$
Let $A$ be the area to be found.

*

*Think about an equilateral triangle with a side length $s$ and $P = 3s \Rightarrow s = \frac{P}{3}$.
Hence $A = \frac{\sqrt{3}}{4} \times (\frac{P}{3})^{2} \approx 0.0481P^{2}$


*Now think about a square. Using the same argument as above $s = \frac{P}{4}$.
Hence $A = 1 \times (\frac{P}{4})^{2} = 0.0625P^{2}$


*Do the same for a regular pentagon to get $s = \frac{P}{5}$.
Hence $A = \frac{\sqrt{5(5 + 2\sqrt{5})}}{4} \times (\frac{P}{5})^{2} \approx 0.0688P^{2}$


*As $s \to \frac{P}{\infty}$, we get a circle.
We can also see the value of $A$ getting larger and larger as the number of sides of the shape increases. This should serve as a rough starting point to understanding why circles enclose the largest area and beyond.
Note: This post by no means highlights a rigorous proof to the asked question. In fact, it does not even prove that $A$ becomes larger and larger as the number of sides increase. Moreover, it fails to consider shapes other than regular polygons. It is simply an observational analysis that should serve as a starting point in exploring the why behind this interesting result.
