Of course you will want to do it differently, but here is something I wrote to play with this. I used the ntheory module because it has different RNGs as well as gcd and Pi.

#!/usr/bin/env perl
use warnings;
use strict;
use ntheory ":all";

cesaro("drand48", sub { int(rand(1<<32)) } );
cesaro("ChaCha20", sub { irand64 } );
cesaro("/dev/urandom", sub { unpack("Q",random_bytes(8)) } );

sub cesaro {
  my($name, $rng) = @_;

  print "Using $name:\n";
  for my $e (1..7) {
    my $n = 10**$e;
    my $t = 0;
    for (1..$n) { $t++ if gcd( $rng->(), $rng->() ) == 1; }
    printf "%8d  %10.8f\n", $n, sqrt(6*$n/$t);
  }
  print "       Pi ", Pi(), "\n\n";
}
[download]

I saved cut-n-paste by passing in the RNG code. One for Perl's default rand (which is drand48 on modern Perl), one for ntheory's CSPRNG, and one that gets data from /dev/urandom. You could replace that with something that got data from random.org. There are even a couple modules that do it automatically (Math::RandomOrg and Data::Entropy). Since we expect t/n ~ 6/Pi^2, a little algebra gives us Pi ~ sqrt(6n/t). I don't believe we can read anything into what the results might mean for the different RNG methods. The results will be different every time unless we seed (and the last can't be seeded since it is O/S collected entropy (long technical discussion of /dev/random vs. /dev/urandom vs. hardware could be had here)).

Using drand48:
      10  2.73861279
     100  3.06186218
    1000  3.15964572
   10000  3.14632429
  100000  3.13988122
 1000000  3.14035598
10000000  3.14143144
       Pi 3.14159265358979

Using ChaCha20:
      10  3.16227766
     100  3.18896402
    1000  3.17287158
   10000  3.15675816
  100000  3.14329189
 1000000  3.13978836
10000000  3.14138726
       Pi 3.14159265358979

Using /dev/urandom:
      10  3.46410162
     100  3.03821810
    1000  3.11588476
   10000  3.14217962
  100000  3.13643021
 1000000  3.14020372
10000000  3.14130744
       Pi 3.14159265358979
[download]

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