So, I had a question for all of you, regarding the relationships between area and circumference, and surface area and volume. For the longest time I was confused as to how these two quantities were related. I saw that the derivative of the area of a circle was equal to its volume, but then that relationship didn't hold true for any other shape. I eventually came to the conclusion that the problem was that you had to measure your area with respect to the "radius" - the line from the center of mass of the shape to the edge. If you do that, you get back an expression which, when differentiated gives you circumference. Here's my work if you'd like to see; its for the simplest case of any n-sided regular polygon, but the essence of the calculations is the same for more complex cases. You just don't get the nice neat formulas at the end :P http://home.comcast.net/~macht/problem.jpg [Broken] My question for this case is, why is the derivative of the area the circumference here. I have a hunch that its due to the fact that you're defining the circumference as a function in polar coordinates and thene finding the area contained within that function, so the derivative would take you back out to the circumference function. However, I couldn't really find a solid way of proving/showing that. However, the second part of this is much more interesting. See, if you take the volume of a sphere and differentiate it, you get the surface area of the sphere. My initial problem was trying to work this out; the 2d version seemed to be a simpler way to approach the problem. However, I soon ran into a problem here. I tried to get the relationship to work for an ellipse, and failed miserably. I found the area of an ellipse with dimensions a, b, and c, to be 4/3 (abc). (Actually, it was kind of funny. I took up about two whiteboards muddling my way through an ugly, ugly triple integration with wierd square root bounds and huge expressions, and at the end it just reduced to that. Definatly one of those "Oh, duh!" moments :P). I tried expressing each in terms of the other, like, x, 2x, and 4x, but still to no avail. I have a feeling I'm missing something, but I can't seem to find a proper equation that defines the surface in spherical coordinates. And I know that the ellipse equations are ugly; this all started from trying to find a quicker method of calculating it. So I guess my question is twofold. First, can any of you confirm my hunch that the results in the 2d case extend to the 3d case? And second, can anyone give me some pointers in attempts to apply it to an ellipse?