C: 131 chars
(Based on the C++ solution by Joey)
main(i,j,c,n){for(scanf("%d",&n),c=0,i|=-n;i<n;puts(""),i+=2)for(j=-n;++j<n;putchar(i*i+j*j<n*n?c++,42:32));printf("%g",2.*c/n/n);}
(Change the i|=-n
to i-=n
to remove the support of odd number cases. This merely reduces char count to 130.)
As a circle:
main(i,j,
c,n){for(scanf(
"%d",&n),c=0,i=1|
-n;i<n;puts(""),i+=
0x2)for(j=-n;++j<n;
putchar(i*i+j*j<n*n
?c++,0x02a:0x020));
printf("%g",2.*c/
n/n);3.1415926;
5358979;}