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

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

and the Volume of the upper segment is

$$V_{\rm seg}={\textstyle{1\over 6}}\pi h(3R^2+h^2),$$ (2)

where

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

$$h = r-{\textstyle{1\over 2}}L,$$ (4)

so the Volume removed upon drilling of a Cylindrical hole is

$$V_{\rm rem} = V_{\rm cyl}+2V_{\rm seg}=\pi[LR^2+{\textstyle{1\over 3}} h(3R^2+h^2)]$$

$$= \pi(LR^2+hR^2+{\textstyle{1\over 3}} h^3)$$

$$= \pi[L(r^2-{\textstyle{1\over 4}}L^2)+(r-{\textstyle{1\over 2}}L)(... ...\textstyle{1\over 4}}L^2)+{\textstyle{1\over 3}} (r-{\textstyle{1\over 2}}L)^3]$$

$$= \pi[Lr^2-{\textstyle{1\over 4}}L^3+(r^3-{\textstyle{1\over 2}}r^2L-{\textstyle{1\over 4}}RL^2+{\textstyle{1\over 8}} L^3)$$

$$\phantom{=} +{\textstyle{1\over 3}}(r^3-{\textstyle{3\over 2}} r^2L+{\textstyle{3\over 4}}rL^2-{\textstyle{1\over 8}}L^3)]$$

$$= \pi[{\textstyle{4\over 3}} r^3+(1-{\textstyle{1\over 2}}-{\textst... ...^2+L^3(-{\textstyle{1\over 4}}+{\textstyle{1\over 8}}-{\textstyle{1\over 24}})]$$

$$= {\textstyle{4\over 3}} \pi r^3-{\textstyle{1\over 6}} \pi L^3={\textstyle{1\over 6}} \pi (8r^3-L^3),$$ (5)

so

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