Simple numerical methods for calculating the digits of $\pi$ Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that can be computed by hand in order to compute the first few digits?

There is also the option of approximating $\pi$ using Monte Carlo integration. 
The idea is this: If we agree that the area of a circle is $\pi r^2$, for simplicity we build a circle of area $\pi$ by setting $r=1$.  Placing this circle wholly inside of another region of known area, preferably by inscribing it in a square of side length 2, then we have a ratio of the circle's area to the total area of the square  (in this example that ratio is $\frac{\pi}{4}$).  
The Monte Carlo method works by approximating areas based on the ratio of the number of sample points lying within our region of interest and the total number of sample points we choose to try.  If we spread a uniformly distributed sequence of $N$ points over our square of area 4, and call the number of points that land inside of the inscribed circle $p$, then we can say $\frac{\pi}{4} = \frac{p}{N}$.  
My implementation of this in Matlab requires tens of thousands of test points to achieve 3.14159xxxx, but I have not tried it for low-discrepancy sequences, or any other uniformly distributed point sets.    
