The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$ and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$
both attain maximum values for finite $n$. We can see from the Wikipedia page on Unit sphere that (for integer $n$ at least), those values are 5 and 7, respectively.
Now, I know that in a strict sense, these values are not really comparable, because they represent measures in different dimensions. But I was wondering if there is nonetheless some good intuitive reason why this happens, both why they both decrease eventually, and why the values where they attain their maximums are those (apparently the actual maxima are fractional, from WolframAlpha, though it didn't give a closed form).