The rate of change of area of a circle with respect to change in its radius is equal to its circumference, as shown below: $$\frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r$$
The rate of change of volume of a sphere with respect to change in its radius is equal to its surface area: $$\frac{dV}{dr} = \frac{dV}{dr}(\frac{4}{3}\pi r^3) = 4\pi r^2$$
Is there a geometrical/mathematical explanation for this?
