A cylinder that circumscribers a hemisphere will share a base with the hemisphere. If the base has radius [tex]r[/tex], then the volume of the hemisphere is
[tex]V_{\rm hemisphere}=\dfrac12\left(\dfrac43\pi r^3\right)=\dfrac23\pi r^3[/tex]
A cylinder with radius [tex]r[/tex] and height [tex]h[/tex] has volume [tex]\pi r^3h[/tex], but a cylinder that circumscribes a hemisphere will have a height equal to the radius of the hemisphere, so [tex]h=r[/tex]. Such a cylinder would have a volume of
[tex]V_{\rm cylinder}=\pi r^3[/tex]
So, the volume of the hemisphere is [tex]\dfrac23[/tex] of the volume of the cylinder.
