1. Coincidence... I think not?!

I've noticed that the circumference of a circle is the derivative of its area. I've also noticed that the surface area of a sphere is the derivative of its volume. Why is this true? Can this correlation be extrapolated to hyperspheres and all n-spheres?

2. What is the analogue of surface area and volume for a hypersphere? What about for n-spheres? Can you determine an explicit formula for those in terms of n? That seems to be the easiest way to demonstrate that one is the derivative of the other.

3. Well, I guess my main question was "why is it true." Let's not go into more than 3 dimensions for now. I'm just confused about what it really means when the rate of change of the area of a circle is the length of its circumference.

4. Well, the rate of change is defined in terms of the radius of the circle. To better understand the connection, can you see why $2 \pi r dr$ approximates the area of a shell around a circle?

5. Originally Posted by icemanfan
Well, the rate of change is defined in terms of the radius of the circle. To better understand the connection, can you see why $2 \pi r dr$ approximates the area of a shell around a circle?
Yes...

6. Originally Posted by mathemagister
Yes...
But how does that explain why the derivative of the highest dimensional measurement of a circle is equal to the 2nd highest dimensional measurement?