Pi Estimation and Visualization

Estimation:

How we know the area of a circle formula

To estimate $\pi$, we must first understand how we know the formula of a circle to be a constant ($\pi$) multiplied by the radius squared. Archimedes used integral calculus to find that when we cut a circle into infinite slices and put them side by side, we get a perfect rectangle.

For more info:

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With this rectangle, we find that the length is half the circumference, and the width is the radius. Using the area of a rectangle formula, when we multiply half the circumference by the radius we get the area of a circle: $$\frac{2 \pi r}{2} \times r = \pi r^2 $$ But when estimating $\pi$, we must pretend we don't know $\pi$. So, all we know is that the area of a circle is equal to a constant $\times r^2$.

Estimating pi

In this program, we estimate $\pi$ in a very similar way Archimedes did, using a regular polygon inscribed inside a circle. With this polygon with $n$ sides, we can measure the apothem of each triangle using the following formula:

Where $A$ is the inside angle formed from the two legs of the triangle with length radius and $a$ is the length of the apothem.

$$\begin{split} \cos A = \frac{a}{r} \\ r\cos A = a \end{split} $$

Therefore, the apothem of each triangle in the polygon, is equal to $r\cos A$.

With the apothem, we can calculate the area of each triangle using $\frac {bh}{2}$ where the base of each triangle is the side length of the polygon and the height is the apothem. With the area of each triangle, we can multiply this my $n$ to get the exact area of the polygon.

Finally, with the area of the polygon, we can input that as the area of the circle into our area formula which is a constant $\times r^2$.

$$ \begin{split} Area_{circle} \approx Area_{polygon} \\ Area_{polygon} = constant \times r^2 \end{split} $$

Knowing the area of the polygon and the radius, we can solve for the unknown constant which would be an approximation of $\pi$.

Visualization

To make the visualization, I used the Turtle graphics module. For the controls, I used the listen method and everytime the user inputted a key or mouse press, the program performs all the calculations and re-renders the drawing again.

Benchmark/Testing

Things to Note:

• The actual circle area remains the same throughout
• All of these tests were performed in as similar of an environment to one another. Meaning, I ran the same background apps (Google Chrome, etc.) and held the tests on the same computer with the same specs ONE AT A TIME.
• My PC specs are as follows:

CPU	GPU	RAM	OS	Python Version	IDE
11th Gen Intel i5-11400 @ 2.6GHz	NVIDIA GeForce RTX 3060	16GB	Windows 11	Python 3.11	PyCharm Community Version 2024.2.0.1

• Render duration increases at a fairly consistent and linear rate
• The 96-sided polygon is the last polygon Archimedes approximating $\pi$ on (granted his method was slightly more complicated and accurate than what this program replicates).
• The first time we reach $\pi$ accurate to 5 decimals places (3.14159) is with a polygon with just below 5000 sides.

Results:

Number of Sides	Actual Circle Area	Polygon Area	Pi Estimate	Pi Error	Render Duration
3	49.483152130933064	20.461118704751872	1.2990381056766584	1.8425545479131347	0.9379889965057373
4	49.483152130933064	31.501953045624983	1.9999999999999987	1.1415926535897944	1.0251493453979492
5	49.483152130933064	37.45017215008198	2.3776412907378814	0.7639513628519117	1.0590147972106934
6	49.483152130933064	40.92223740950367	2.5980762113533125	0.5435164422364807	1.1340358257293701
7	49.483152130933064	43.1011326380236	2.7364101886380974	0.40518246495169574	1.1997427940368652
8	49.483152130933064	44.55048923836323	2.828427124746185	0.31316552884360815	1.2585244178771973
9	49.483152130933064	45.560396471973434	2.8925442435894224	0.2490484100003707	1.334967851638794
10	49.483152130933064	46.29095854657074	2.9389262614623597	0.20266639212743343	1.3704252243041992
20	49.483152130933064	48.67319423549873	3.09016994374947	0.051422709840323044	2.064829111099243
50	49.483152130933064	49.353020484988214	3.1333308391076007	0.008261814482192431	5.422649145126343
75	49.483152130933064	49.42529046775846	3.1379191249618508	0.0036735286279423462	5.596475839614868
96	49.483152130933064	49.44783134507798	3.1393502030468747	0.002242450542918384	6.798015832901001
100	49.483152130933064	49.450599948071165	3.1395259764656953	0.0020666771240978044	7.013974905014038
250	49.483152130933064	49.47794291800626	3.141261930417217	0.00033072317257598627	16.565859079360962
500	49.483152130933064	49.48184979686067	3.1415099708386314	0.0000826827511617445	32.386380672454834
1000	49.483152130933064	49.48282654549867	3.141571982780341	0.000020670809452116856	63.49086785316467
2500	49.483152130933064	49.483100037209645	3.141589346256858	0.000003307332935076346	158.71818161010742
5000	49.483152130933064	49.48313910744267	3.1415918267527796	0.0000008268370135233738	314.69306540489197

Graphs:

Relation of the number of sides on the polygon to the $\pi$ error

Relation of the number of sides on the polygon to the render duration