Approximating is always fun. Some approximations are well known, like Zu Chongzhi’s, , that is fairly precise (it gives 6 digits).
We can derive a good approximation using continue fractions, series, or some spigot algorithm. We can also use Monte Carlo methods, such as drawing uniformly distributed values inside a square, count those who falls in the circle, then use the ratio of inside points to outside points to evaluate . This converges slowly. How do we evaluate the area of the circle then? Well, we could do it exhaustively, but in a smart way.