Monte Carlo estimate of pi and its error

How can you estimate $\pi$ with a Monte Carlo simulation, and how does the error shrink with the number of samples?

Draw $(x,y)$ uniformly in the unit square and multiply the fraction with $x^2+y^2\le1$ by $4$. The quarter disc has area $\pi/4$, so each point is a Bernoulli trial with that success probability. The estimator's standard deviation is $\sqrt{\pi(4-\pi)/n}\approx1.64/\sqrt n$, so error falls like $n^{-1/2}$.