I had a question from the course of school stereometry: why is the volume of a ball calculated exactly according to the formula $V=\frac{4}{3} \pi R^3$, why is $4/3$ used?
I did not find the necessary information in the textbook and on the Internet. I will be glad to have a good explanation.
Below, see a half ball, a cone, and a cyclinder (all sideways and therefore looking like a half disk, a triangle, a rectangle). The all have radius $r$ and height $r$. Look at the plane at heihgt $h$ above the ground (dotted line). The intersection with the cylinder is of course a circle of radius $r$ and area $\pi r^2$; the intersection with th econe is a circle of radius $h$ and area $\pi h^2$; the intersection with the half ball is a circle of radius $\sqrt{r^2-h^2}$ (use Pythagoras to find the radius) and area $\pi(r^2-h^2)$. Hence - independently of $h$ - the coross section area of the half ball plus the cross section area of the cone equals the cross section area of the cylinder: $$\pi(r^2-h^2)+\pi h^2=\pi r^2. $$
Convince yourself that this means that the same equation holds for the volumes: $$ V_{\text{half sphere}} + V_{\text{cone}} = V_{\text{cylinder}}.$$ You certainly know that the volume of the cylinder is $\pi r^3$ and that the volume of the cone is $\frac 13\pi r^3$ (where the factor $\frac13$ comes in exactly the same way as it does for pyramids). Therefore, the volume of the half sphere is $\frac23\pi r^3$ and the volume of the full sphere is $$ \frac43\pi r^3.$$
