monte carlo simulation measured from the outside of the wall and the diameter from the insideor perhaps it' just poetic license we leave it to the reader to decide archimedes of syracuse ( - bcderived upper and lower bounds on the value of by using high-degree polygon to approximate circular shape using polygon with sideshe concluded that / / giving upper and lower bounds was rather sophisticated approach for the time alsoif we take his best estimate as the average of his two bounds we obtain an error of about not badlong before computers were inventedthe french mathematicians buffon ( and laplace ( - proposed using stochastic simulation to estimate the value of think about inscribing circle in square with sides of length so that the radiusrof the circle is of length by the definition of parea pr since is area but what' the area of the circlebuffon suggested that he could estimate the area of circle by dropping large number of needles (which he argued would follow random path as they fellin the vicinity of the square the ratio of the number of needles with tips lying within the square to the number of needles with tips lying within the circle could then be used to estimate the area of the circle if the locations of the needles are truly randomwe know that!"#!!"#!$!""#$"!!"#!$!"#!!"#$%!""#$"!!"#$%solving for the area of the circle!"#!!"#!$!"#!!"#$%!""#$"!!"#!$!""#$"!!"#$%recall that the area of by square is so!"#!!"#!$ !""#$"!!"#!$!""#$"!!"#$%in generalto estimate the area of some region pick an enclosing regionesuch that the area of is easy to calculate and lies completely within pick set of random points that lie within let be the fraction of the points that fall within multiply the area of by buffon proposed the idea firstbut there was an error in his formulation that was later corrected by laplace