that

Area of Circle = f(r), and Circumference of Circle = f'(r)

AND

volume of sphere = g(r) and surface area of sphere = g'(r)

is this a coincidence?

Ofcourse not!

You get a sphere by integrating many surfaces and a circle by integrating many circumferences

]]>Consider the change in volume of the volume of a sphere dv coresponding to dr, this should be like adding a thin coating of paint onto the surface, increasing its volume ever so slightly. The total area of this coating would be the surface area, and the thickness would be dr.

Likewise, consider a circle thats getting wider. The slight change in the surface area should be like wrapping a string once around the outside to increase the area. The width of the string would be dr, and the length would be the circumference of the circle.

Now, who wants to explain why this makes sense in 4 dimensions?

]]>Given "hypercube" with side of length > 1, it's area diverges as you go up in dimensions.

Given a hypersphere with radius of any length, it's area goes to 0 as you go up in dimensions.

Any more like that I wonder? Or is it special to spheres?

]]>but I know nothing about it really.]]>

Area of Circle = f(r), and Circumference of Circle = f'(r)

AND

volume of sphere = g(r) and surface area of sphere = g'(r)

is this a coincidence?

]]>