Natural Pi #0 - Rock
APL, 23 bytes

{.5*⍨6×⍵÷1+.=∨/?⍵2⍴1e6}


Explanation:


?⍵2⍴1e6: generate a 2-by-⍵ matrix of random numbers in the range [1..106]
1+.=∨/: get the GCD of each pair and see how many are equal to 1. This calculates S. 
.5*⍨6×⍵÷: (6 × ⍵ ÷ S)0.5