Why is the volume of a sphere $\frac{4}{3}\pi r^3$? I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for the formula?

Pappus's centroid theorem (second theorem) says that the volume of a solid formed by revolving a region about an axis is the product of the area of the region and the distance traveled by the centroid of the region when it is revolved.  A sphere can be formed by revolving a semicircle about is diameter edge.
The area of the semicircle is $\frac{1}{2}\pi r^2$.  The centroid of the semicircle can be found by intersecting two lines that both divide the area of the semicircle into two equal parts.  One such line is perpendicular to the diameter edge through the center of the semicircle (this is a line of symmetry of the semicircle).  Another such line is parallel to the diameter edge, $\frac{4r}{3\pi}$ away from it (verification of this is left as an exercise for the reader).  When revolved about the diameter edge of the semicircle, the centroid travels $2\pi\cdot\frac{4r}{3\pi} = \frac{8}{3}\cdot r$, so the volume of the sphere is $\frac{1}{2}\pi r^2\cdot\frac{8}{3}\cdot r = \frac{4}{3}\pi r^3$.
