\begin{frame} \frametitle{Motivation and Overview} But what did the ancient Greek do with curved figures? \begin{center} \begin{tikzpicture}[default,scale=.7] \onslide<1,2>{ \draw (0,0) circle (1cm); } \foreach \c in {2,3,4,5,6,7} { \setcounter{tmpcounter}{\c} \addtocounter{tmpcounter}{1} \onslide<\arabic{tmpcounter}->{ \begin{scope}[xshift={(\c-2)*2.5cm}] \draw (0,0) circle (1cm); \inscribe{\c} \end{scope} } } \end{tikzpicture} \end{center} \pause They inscribed polygons into the figure: \begin{itemize} \foreach \c in {3,4,5} { \pause \pgfmathparse{1/2*\c*sin(360/\c)} \item first a polygon with $\c$ points $\implies$ area $A_\c = \pgfmathresult$, } \pause\pause\pause\pause \item \ldots \foreach \c in {10,100} { \pgfmathparse{1/2*\c*sin(360/\c)} $A_{\c} = \pgfmathresult$, }\ldots \end{itemize} \pause\medskip If you continue, you will see: \begin{itemize} \pause \item the values get closer and closer to $\pi = 3.141 592 653 \ldots$ \end{itemize} \pause \begin{block}{} Area $A$ of the circle is the \emph{limit} of the sequence $A_3,A_4,A_5,\ldots$ \pause \begin{talign} \text{A} = \lim_{n\to\infty} A_n \end{talign} \end{block} \end{frame}