Calculate π By Throwing Darts

We can approximate π using nothing more than random numbers and some simple geometry: we randomly throw darts at a square board of side r; within the square we inscribe a quadrant of a circle of radius r with its centre at (0, 0). We count all of the 'throws'; if a dart lands within the quadrant, we also count a 'hit'.

For a large number of throws, we see that:

hits throws = area-of-quadrant area-of-square

Some half-remembered geometry tells us that:

area-of-quadrant area-of-square = πr2/4 r2 = π 4


π = 4 ( area-of-quadrant area-of-square ) = 4 ( hits throws )

I first solved this problem as an undergraduate sometime in 1994 as part of a Computational Physics module. Using FORTRAN.

#!/usr/bin/env python

import random
import math

# class representing a single throw of a dart
class Throw:
	def __init__(self):
		# generate two random coordinates and work out how far away we are from
		# the origin
		self.x = random.random()
		self.y = random.random()
		self.distance = self.distance()

	def distance(self):
		# the distance from the origin is the hypotenuse of a right-angled
		# triangle with sides of length and x and y. Pythagoras told us that:
		#    distance = sqrt((x^2) + (y^2))
		# which looks like this in python
		return math.sqrt(self.x**2 + self.y**2)

	# did we land inside the quadrant?
	def is_a_hit(self):
		return self.distance <= 1.0

# main class
class MonteCarlo:
	def __init__(self):
		# initialise everything
		self.hits = 0
		self.throws = 0
		self.pi = 0

	# this method is called on every throw
	def increment(self, throw):
		# we always clock up another throw
		self.throws += 1
		# and accumulate a hit if we scored
		if throw.is_a_hit():
			self.hits += 1
		# then get a new value of pi

	# explanation can be found here:
	def calculate_pi(self):
		self.pi = 4 * (float(self.hits) / float(self.throws))

	# we use this in determining whether to print our status
	def divides_by(self, number):
		return float(self.throws) % number == 0

	# represent the current state as a string
	def __repr__(self):
		return "Throws: %10d, Hits: %10d, Pi: %10f, Actual Pi: %10f" % (self.throws, self.hits, self.pi, math.pi)

# if we're called on the command line
if __name__ == '__main__':
	import sys
		step = int(sys.argv[1])
	except IndexError:
		step = 1000
	# construct a new instance
	m = MonteCarlo()
	# loop forever
	while 1:
		# keep throwing darts
		# only print on every nth iteration
		if m.divides_by(step): print m

There's now a ruby version, too.

Published under a Creative Commons Attribution-NonCommercial licence