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In this article I will show how you can use a monte carlo simulation to estimate the value of pi.
Imagine a circle with a radius of one that fits perfectly inside of a square. This square has a side length of two, twice the radius of the circle. The following graphic visualizes this where the circle is in green and the square in red.
From our basic geometry knowledge, we know the following about the area of the shapes:
\[r := \text{the radius of the circle} = 1 \\ A_c := \text{area of the circle} = \pi * r ^ 2 = \pi \\ A_s := \text{area of the square} = (2r) ^ 2 = 4\]Therefore, we know the following about the ratio of the sizes of the shapes.
\[\frac{A_c}{A_s} = \frac{\pi}{4}\]This is all well and good, but we want to actually estimate the value of pi. To do this, we use a monte carlo simulation.
For the simulation, we will iteratively place points somewhere randomly within the square. We will record the number of points that are placed and the number of points that are within the circle. By doing so, we would expect the ratio of the number of points that fall within the circle to the total number of points to be equal to the ratio of the area of the circle to the area of the square.
That sets up the following:
\[C := \text{set of points within the circle} \\ S := \text{set of points within the square} \\ \frac{|C|}{|S|} \approx \frac{A_c}{A_s} = \frac{\pi}{4} \\ \pi \approx \frac{|C| * 4}{|S|}\]Now all that’s left for us is to run the simulation. Below you will see exactly that. The points that fall within the circle are in green and the points outside of the circle are in red. As we add more and more points, we expect our estimate of pi to improve.