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\begin{document}
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\Huge\sf \textbf{Some {\green fascinating} properties of $\pi$
{\orange(check them out!)}}
\end{center}

\bigskip

\LARGE\sf
%\begin{center}
\textbf{Definition:} In {\red any} circle the ratio $\frac{\text{\blue circumference}}{\text{\blue diameter}}=\frac{C}{d}$ is a {\red constant}
called \ $\pi$

$\Longrightarrow\quad C=\pi d=2\pi r$ where $r=\frac{d}{2}$ is the {\blue radius}

From {\green trigonometry} \ $%\sin{\pi\over2}=1=
\tan{\pi\over4}=1$\quad$\Longrightarrow$\quad$%2\sin^{-1}(1)=
{\pi\over4}=\tan^{-1}(1)$
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\end{pspicture}\qquad\qquad

From {\green calculus} \ $\tan^{-1}(x)=\int{dx\over x^2+1}=\int
1-x^2+x^4-x^6+\cdots dx=x-{x^3\over3}+{x^5\over5}-{x^7\over7}+\cdots$

$\Longrightarrow\quad \pi=4-{4\over3}+{4\over5}-{4\over7}+{4\over9}-{4\over11}+{4\over13}-\cdots$ {\purple(Gregory-Leibniz$\sim$1675)}
$=$ \decreasingdigits{30}{3}{14159265358979323846264338327950288419}

$\pi=6\tan^{-1}\big({\sqrt3\over3}\big)=2\sqrt3\big(1-{1\over3^13}+{1\over3^25}-{1\over3^37}+{1\over3^49}-\cdots\big)$ {\purple(Sharp 1699)}
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{\blue Area} of a circle of radius $r$ is\quad $\displaystyle 4\int_0^r\sqrt{r^2-x^2}dx=\ldots=\pi r^2$ \quad {\orange
(substitute $x=r\sin\theta$)}

{\blue Volume} of a sphere of radius $r$ is\quad
${\displaystyle 2\int_0^r\pi(r^2-x^2)\,dx}=\ldots=\frac{4}{3}\pi r^3=\frac{1}{6}\pi d^3$ \quad {\orange(easy)}

%${\displaystyle 2\int_0^r2\pi x\sqrt{r^2-x^2}\,dx}=\ldots=\frac{4}{3}\pi r^3$

$e^{i\pi}+1=0$\quad {\purple(Euler 1735)} \quad
$%{\displaystyle\sum_{n=1}^\infty}{1\over n^2}=
1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{49}+\frac{1}{64}+\dots=\frac{\pi^2}{6}$\quad{\orange(try it)}


%\end{center}

\bigskip

\begin{center}\tiny 
Fascinating properties of $\pi$ --- 
Math Poster 2007 --- math.chapman.edu
\end{center}
\end{document}