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Spherical Ring


A Sphere with a Cylindrical Hole cut so that the centers of the Cylinder and Sphere coincide, also called a Napkin Ring. The volume of the entire Cylinder is

V_{\rm cyl}=\pi LR^2,
\end{displaymath} (1)

and the Volume of the upper segment is
V_{\rm seg}={\textstyle{1\over 6}}\pi h(3R^2+h^2),
\end{displaymath} (2)

$\displaystyle R$ $\textstyle =$ $\displaystyle \sqrt{r^2-{\textstyle{1\over 4}}L^2}$ (3)
$\displaystyle h$ $\textstyle =$ $\displaystyle r-{\textstyle{1\over 2}}L,$ (4)

so the Volume removed upon drilling of a Cylindrical hole is

$\displaystyle V_{\rm rem}$ $\textstyle =$ $\displaystyle V_{\rm cyl}+2V_{\rm seg}=\pi[LR^2+{\textstyle{1\over 3}} h(3R^2+h^2)]$  
  $\textstyle =$ $\displaystyle \pi(LR^2+hR^2+{\textstyle{1\over 3}} h^3)$  
  $\textstyle =$ $\displaystyle \pi[L(r^2-{\textstyle{1\over 4}}L^2)+(r-{\textstyle{1\over 2}}L)(...
...\textstyle{1\over 4}}L^2)+{\textstyle{1\over 3}} (r-{\textstyle{1\over 2}}L)^3]$  
  $\textstyle =$ $\displaystyle \pi[Lr^2-{\textstyle{1\over 4}}L^3+(r^3-{\textstyle{1\over 2}}r^2L-{\textstyle{1\over 4}}RL^2+{\textstyle{1\over 8}} L^3)$  
  $\textstyle \phantom{=}$ $\displaystyle +{\textstyle{1\over 3}}(r^3-{\textstyle{3\over 2}} r^2L+{\textstyle{3\over 4}}rL^2-{\textstyle{1\over 8}}L^3)]$  
  $\textstyle =$ $\displaystyle \pi[{\textstyle{4\over 3}} r^3+(1-{\textstyle{1\over 2}}-{\textst...
...^2+L^3(-{\textstyle{1\over 4}}+{\textstyle{1\over 8}}-{\textstyle{1\over 24}})]$  
  $\textstyle =$ $\displaystyle {\textstyle{4\over 3}} \pi r^3-{\textstyle{1\over 6}} \pi L^3={\textstyle{1\over 6}} \pi (8r^3-L^3),$ (5)

V_{\rm left} = V_{\rm sphere}-V_{\rm rem} = {\textstyle{4\ov...
...xtstyle{1\over 6}} \pi L^3)
= {\textstyle{1\over 6}} \pi L^3.
\end{displaymath} (6)

© 1996-9 Eric W. Weisstein