4718.sagews

### This script estimates pi by shooting randomly at a circle inscribed in a square

var ('x y r n')
r = 1
n = 100
points = []
inside = 0

### Shoot randomly into the square:
for i in range(0,n):
    [x,y]=[random(),random()]
    points.append([x,y])
    
### If a shot lands inside the circle, make a note of it
    if (y <= sqrt((r^2)-(x^2))):
        inside += 1
        
### Approximate pi based on the fraction of shots that landed in the circle
### Area of circle = pi*r^2
### Area of square = (2*r)^2 = 4*r^2
### Shots in circle / Shots in square = (pi*r^2)/(4*r^20 = pi()/4

piapprox = 4*(inside / n)
estimate = "Based on "
estimate += str(n)
estimate += " shots, pi is ~= "
estimate += str(piapprox.n())

show(estimate)

### Graph the solution
circle = []
for i in range(0,1000):
    x = i/1000
    y = sqrt((r^2)-(i/1000)^2)
    circle.append([x,y])

graph = list_plot(points)
graph += list_plot(circle,color='red',figsize=[6,4],plotjoined=true)
show(graph)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|Based|\phantom{\verb!x!}\verb|on|\phantom{\verb!x!}\verb|100|\phantom{\verb!x!}\verb|shots,|\phantom{\verb!x!}\verb|pi|\phantom{\verb!x!}\verb|is|\phantom{\verb!x!}\verb|~=|\phantom{\verb!x!}\verb|3.04000000000000|$$