consider:
a square, of side r, with an arc drawn from (r,0) to (0,r) around (0,0).
As - Area of square
Aa - Area inside of arc
Ds - "darts" falling in square (total darts)
Da - "darts" falling in arc (somewhat less than Ds)
k - a constant:
relating to the density of "darts" distributed uniformly
(we suppose) throughout the area Ds.
Pi - the irrational number
2
2 (Pi)kr
Ds = kr Da = ------
4
Da Pi Da
-- = -- ... 4 -- = Pi
Ds 4 Ds
The number of darts falling within r of the origin is noted in Da,
the total number of darts fallen is noted in Ds. The ratio is assumed to
approach Pi as Ds approaches infinity (oo).