#!/usr/bin/python3
 
from math import sin, cos, atan2, pi
import random
 
rnd = random.Random(1)
 
def rndpt(a):
    while True:
        x = rnd.uniform(-1, 1)
        y = rnd.uniform(-1, 1)
        if x*x + y*y > 1:
            continue
        ang = atan2(y, x)
        if ang < 0:
            ang += 2*pi
        if ang > a:
            continue
        return (x, y)
 
def trial(a, rep=10000):
    asum = 0
    for iteration in range(rep):
        x1, y1 = rndpt(a)
        x2, y2 = rndpt(a)
        asum += abs(x1*y2 - x2*y1)
    return (asum/rep/2, (4*(a-sin(a)))/(9*a*a))
 
for n in range(1, 8):
    r, c = trial(2*pi/n)
    print("2*pi/{}: {:+.6f} = {:.6f} - {:.6f}".format(n, r-c, r, c))
 

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