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\begin{document}
 
\noindent
Eric Rasmusen\\
Feb. 5, 2025
\begin{center}
{\large {\bf  Formulas for  $\pi$}}
\end{center}
 
This list is selected from the {\it Wikipedia}  article, \href{https://en.wikipedia.org/wiki/List_of_formulae_involving_\%CF\%80}{ ``List of Formulas involving $\pi$''. } 
See also 
  Peter Borwein,    \href{http://www.pi314.net/ref/P159.pdf} {``The Amazing Number $\Pi$''} (2000). 
%I really should make these equnarrays to line up. 

\begin{eqnarray} \label{e1}
 \pi  &=  &\frac{C}{d}      \hspace{36pt}   {\rm ( where \  C\ is\  the  \ circumference \  and \ d  \ is  \ the  \ diameter \  of \  a  \ circle.        ) } \\
   \notag &&\\
 \label{e2}
 \pi& =   &\frac{C}{2r}     \hspace{36pt}   {\rm (where \  C \  is  \ the  \ circumference   \ and  \  r   \ is  \  the  \  radius   \ of  \  a   \ circle.         ) } 
\end{eqnarray}


 \begin{equation}\label{e2}
 \pi = \frac{A}{r^2}     \hspace{36pt}   {\rm (    where  \ A  \ is  \ the \  area   \ and  \      r is  \  the  \  radius  \  of   \ a  \ circle.    ) }
\end{equation}


  \begin{equation}\label{e2}
 \pi = \frac{A}{\frac{major}{2} \cdot \frac{minor}{2}},
 \end{equation}
where $A$ is    the area of an ellipse,  $major$ is its major axis, and $minor$ is its minor axis. 

 \begin{equation}\label{e2}
 \pi = \frac{3}{4}  \cdot \frac{V}{r^3}      \hspace{36pt}   {\rm (  where  \ V \  is \  the   \ volume \  of  \  a   \ sphere    \   and   \ r   \ is  \  its  \  radius. 
      ) }
 \end{equation}


 \begin{equation}\label{e2}
 \pi =  \frac{A}{4r^2}    \hspace{36pt}   {\rm (   where \  A  \  is \  a   \ sphere's   \ surface  \  area  \  and  \  r  \   is  \  its   \ radius.       ) }
 \end{equation}


 

 \begin{equation}\label{e2} 
  \pi =   
 4\left(  
 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots  \right)   \hspace{36pt}  {\rm  (Madhava, Gregory, Leibniz  \ formula \ for \ } \pi)
  \end{equation}
    


 \begin{equation}\label{e2} 
  \pi =   
\sqrt{  6} \cdot  \sqrt{   1 +  \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} +  \cdots  }   \hspace{36pt}  {\rm  (Euler \ first \ formula  for \   \pi})
  \end{equation}
    
 
 
 
\begin{equation}\label{e2}
       \pi =    \sum_{k=1}^\infty k\frac{2^k k!^2}{(2k)!} - 3 \hspace{36pt}   {\rm  (Euler  \ second \ formula   \ for \ } \pi)
 \end{equation} 
 
 
 

 
  \end{document}