I came across this post which gives a method to estimate Pi by using a circle, it’s circumscribed square and (lots of) random points within said square. Booth used Stata to estimate Pi, but here’s some R code to do the same thing…

x <- 0.5 # center x
y <- 0.5 # center y
n <- 1000 # nr of pts
r <- 0.5 # radius
pts <- seq(0, 2 * pi, length.out = n)
plot(sin(pts), cos(pts), type = 'l', asp = 1) # test

require(sp)
xy <- cbind(x + r * sin(pts), y + r * cos(pts))
sl <- SpatialPolygons(list(Polygons(list(Polygon(xy)), "polygon")))
plot(sl, add=FALSE, col = 'red', axes=T )


# the square
xy <- cbind(c(0, 1, 1, 0), c(0, 0, 1, 1))
sq <- SpatialPolygons(list(Polygons(list(Polygon(xy)), "polygon")))

plot(sq, add = TRUE)

N <- 1e6
x <- runif(N, 0, 1)
y <- runif(N, 0, 1)
sp <- SpatialPoints(cbind(x, y))
plot(sp, add = TRUE, col = "green")

require(rgeos)
(sim_pi <- (sum(gIntersects(sp, sl, byid = TRUE))/N) *4)
sim_pi - pi

Note the use of sp and rgeos packages to calculate the intersections.