Why isn't the volume of a sphere: $\pi$$^\text{2}$$r^\text{3}$, instead it is $\frac{4}{3}$$\pi$$r^\text{3}$? Like wise the surface area is 4$\pi$$r^\text{2}$and not 2$\pi$$^\text{2}$$r^\text{2}$.
Simply take a 2D circle and rotate on same center and radius perpendicular circle and we get a sphere. But this isn't consistent among all the shapes which have a common axis.
I believe the only repeated/common things in this derivation are the pole of intersection and the axis.
Thanks for your help.
Although a sphere can be formed by rotating a 2d full circle by 180 degree, during the rotation every point on the 2d circle will travel by a different amount of distance. For example, the two points that is farthest from the rotation axis will travel by a distance of $\pi r$, the other points will travel by a distance $\pi d$ (where $d$ is the perpendicular distance to the rotation axis and $d < r$) and the two points at the intersection of the 2d circle and the rotation axis will not moved at all (i.e., $d=0$). Therefore, you cannot simply multiply full circle's area $\pi r^2$ and circumference length $2\pi r$ by $\pi r$ to obtain sphere's volume and surface area.