## Monte Carlo Approximation of Pi

Given a unit circle (r=1) inscribed inside a square of side length 2, we can calculate the areas of each shape as follows:

\[A_{square} = (2r)^2 = 4 A_{circle} = \pi (r)^2 = \pi\]

Using a Monte Carlo simulation, we can then assess how many random points land within the circle compared to how many points land outside of the circle.

Proportion of points inside circle = \(\frac{A_{circle}}{A_{square}} = \frac{\pi}{4}\)

Therefore, we approximate \(\pi\) as 4*proportion.

\[A_{square} = (2r)^2 = 4 A_{circle} = \pi (r)^2 = \pi\]

Using a Monte Carlo simulation, we can then assess how many random points land within the circle compared to how many points land outside of the circle.

Proportion of points inside circle = \(\frac{A_{circle}}{A_{square}} = \frac{\pi}{4}\)

Therefore, we approximate \(\pi\) as 4*proportion.

The method is of course fairly straightforward, so I won't belabor you with the details here. The majority of the code is actually for

**pygame**'s visualization rather than the actual approximation.Classic Fizzbuzz

Below is a screenshot of the

**pygame**window. We see that the approximation is already quite good with relatively few points.