The area of a circle is calculated by pi*r^2, and the area of the bounding square is (2r)^2 = 4r^2. Dividing the area of the circle by the area of the square we get the ratio of the two areas:
area of circle / area of square = pir^2 / 4r^2 = pi / 4
This means we can estimate pi using the formula:
pi = simulated points within circle / total number of simulated points
By simulating random numbers within this square, we get approximated values for the area of the circle and the area of the square. Hence, we can estimate
To identify which points fall within the circle we use the equation of the circle: (x-a)^2+(y-b)^2 = r^2, where (a, b) is the circle center
In our example the circle center is (0, 0) and the radius is 1, so simulated points that satisfy sqrt{x^2+y^2} <= 1 are within the circle.