When I was an undergraduate, I attempted to generalize the common observation that $\dfrac{d}{dr} \pi r^{2} = 2\pi r$ for other shapes (that is, when is the derivative of area equal to perimeter for some other shape?).
My finding was that for any regular polygon, differenting the area of a regular polygon with respect to the radius of its inscribed circle will give its perimeter as a function of the radius of its inscribed circle.
I detailed my experience with the problem here (https://youtu.be/0vYWsOBBXxw), and because of the experience I had with it (starting with a common observation, finding a pattern, making a conjecture, and then proving it) I have given a talk on the problem to the undergraduate math majors at my university a couple of times, I think the problem has a lot of pedogogical value at that level.
Anyways, last week, I gave the talk, and one of the professors that attended remarked that he thinks that the general result here is that given any polygon, if we find its "center of mass", then the measurement we want is the shortest distance from the center of mass to the edge of the polygon.
Can anyone comment on what the more general result is here? That is, in describing "when the derivative of area is perimeter", what's the most general statement we can say about the most general shapes? Or, how could we show that the center of mass to edge is always the measurement we want?
