Estimating Pi

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List as many ways as you can think of for estimating the value of $\pi$ .

Draw a (fairly large) circle on a blank sheet of paper. We’ll think of this as a unit circle.

Divide the unit circle into $2^2 = 4$ equal wedges each with its vertex at the center of the circle $O$ . On each wedge, call the two corners of the wedge that lie on the circle $A$ and $B_2$ . Let $\mathcal {A}_2$ denote the area of the triangle $\triangle OAB_2$ and let $\theta _2$ denote the measure of the angle at $O$ . Explain how to estimate the area of the circle with triangle $\triangle OAB_2$ . What is your estimate?

Divide the unit circle into $2^3 = 8$ equal wedges each with its vertex at the center of the circle $O$ . On each wedge, call the two corners of the wedge that lie on the circle $A$ and $B_3$ . Let $\mathcal {A}_3$ denote the area of the triangle $\triangle OAB_3$ and let $\theta _3$ denote the measure of the angle at $O$ . Explain how to estimate the area of the circle with triangle $\triangle OAB_3$ . What information do you need to know to actually do this computation?

Given an angle $\theta$ , explain the relation of $\sin (\theta )$ and $\cos (\theta )$ to the unit circle. How could these values help with the calculation described above?

Divide the unit circle into $2^n$ equal wedges each with its vertex at the center of the circle $O$ . On each wedge, call the two corners of the wedge that lie on the circle $A$ and $B_n$ . Let $\mathcal {A}_n$ denote the area of the triangle $\triangle OAB_n$ and let $\theta _n$ denote the measure of the angle at $O$ . Explain why someone would be interested in the value of:

$\sin \left (\frac {\theta _n}{2}\right )$

Recalling that:

$\sin \left (\frac {\theta }{2}\right ) = \sqrt {\frac {1-\cos (\theta )}{2}} \qquad \text {and}\qquad \cos (\theta )^2 + \sin (\theta )^2 = 1$

Explain why:

$2 \mathcal {A}_{n+1} = \sqrt {\frac {1 - \sqrt {1 - (2\mathcal {A}_n)^2}}{2}}$

Let’s fill out the following table (a calculator will help!):

$\begin{array}{| c || c | c | c | c | c |} \hline n & \mathcal {A}_n & \text {Approx. Area} & \sqrt {1-(2\mathcal {A}_n)^2} & \frac {1 - \sqrt {1-(2\mathcal {A}_n)^2} }{2} &\rule [0mm]{0mm}{6mm}2\mathcal {A}_{n+1} =\sqrt {\frac {1 - \sqrt {1-(2\mathcal {A}_n)^2} }{2}} \\ \hline \hline 2 &\rule [7mm]{20mm}{0mm}\hspace {20mm} &\hspace {20mm} &\hspace {20mm} & \hspace {20mm} & \hspace {20mm}\\ \hline 3 &\rule [0mm]{0mm}{7mm} & & & & \\ \hline 4 &\rule [0mm]{0mm}{7mm} & & & & \\ \hline 5 &\rule [0mm]{0mm}{7mm} & & & & \\ \hline 6 &\rule [0mm]{0mm}{7mm} & & & & \\ \hline 7 &\rule [0mm]{0mm}{7mm} & & & & \\ \hline 8 &\rule [0mm]{0mm}{7mm} & & & & \\ \hline \end{array}$

What do you notice?

2025-01-06 15:56:25

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*	$\cdot$
infinity	$\infty$
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arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
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mu	$\mu$
nu	$\nu$
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pi	$\pi$
rho	$\rho$
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psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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