\(n\) must be an integer for valid solutions to area and volume of the torus. Given \(r_p\) and \(r_{or}\), what is the minimum \(n_z\)? But a trained master will instead, consider
\(V_{ tor }=\cfrac { \pi }{ n } r_{ or }r_{ p }\left\{ (2\pi r_{ or })^{ 2 }+(n2\pi r_{ p })^{ 2 } \right\} ^{ 1/2 }\)
In the limit \(n\rightarrow\infty\),
\(\lim\limits_{n\rightarrow\infty}{V_{ tor }}=\lim\limits_{n\rightarrow\infty}{\cfrac { \pi }{ n } r_{ or }r_{ p }\left\{ (2\pi r_{ or })^{ 2 }+(n2\pi r_{ p })^{ 2 } \right\} ^{ 1/2 }}\)
\(\lim\limits_{n\rightarrow\infty}{V_{ tor }}=\lim\limits_{n\rightarrow\infty}{\pi r_{ or }r_{ p }\left\{ (\cfrac{2\pi r_{ or }}{n})^{ 2 }+(2\pi r_{ p })^{ 2 } \right\} ^{ 1/2 }}\)
\(V_{ tor }={\pi r^2_{ p }.2\pi}r_{or}\)
which is the published result for the volume of a torus. In the limit \(n\rightarrow\infty\) it does not matter whether \(n\) is a integer or not.
In a similar way, the surface area of the torus is given by,
\(A_{tor}=\lambda L_{or}=\cfrac{2\pi r_{or}}{n}\left\{( 2\pi r_{or})^2 +( n2\pi r_{ p } ) ^{ 2 }\right\}^{1/2}\)
In the limit \(n\rightarrow\infty\),
\(\lim\limits_{n\rightarrow\infty}{A_{tor}}=\lim\limits_{n\rightarrow\infty}{\cfrac{2\pi r_{or}}{n}\left\{( 2\pi r_{or})^2 +( n2\pi r_{ p } ) ^{ 2 }\right\}^{1/2}}\)
\(\lim\limits _{ n\rightarrow \infty }{ A_{ tor } } =\lim\limits _{ n\rightarrow \infty }{ 2\pi r_{ or }\left\{ (\cfrac { 2\pi r_{ or } }{ n } )^{ 2 }+(2\pi r_{ p })^{ 2 } \right\} ^{ 1/2 } } \)
\( A_{ tor }=2\pi r_{ or }2\pi r_{ p }\)
which is very nice.
