Pack the sphere (yellow) in a bicylinder (white). Pack the bicylinder in a cube (red). Using a plane parallel to the bicylinder axes, calculate the ratio of the circle intersecting the sphere with the square intersecting the bicylinder: ${\displaystyle \textstyle {\frac {\pi r^{2}}{(2r)^{2}}}={\frac {\pi }{4}}}$.

Next, calculate the bicylinder volume by noting that the difference between the square of intersection for the cube and for the bicylinder is equivalent to squares in each corner. These squares form 8 isosceles pyramids (blue) as the plane is moved across the solids. The volume of the cube (red) minus the volume of the eight pyramids (blue) is the volume of the bicylinder (white). The volume of the 8 pyramids is: ${\displaystyle \textstyle 8\times {\frac {1}{3}}r^{2}\times r={\frac {8}{3}}r^{3}}$, and then we can calculate that the bicylinder volume is ${\displaystyle \textstyle (2r)^{3}-{\frac {8}{3}}r^{3}={\frac {16}{3}}r^{3}}$.

Finally, use the ratio of the bicylinder volume to the sphere volume to calculate the volume of the sphere:

${\displaystyle {\frac {\pi }{4}}\times {\frac {16}{3}}r^{3}={\frac {4\pi }{3}}r^{3}}$.