Wolfram Alpha:

Buffon's Needle
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When we are in school we learn that π is used
whenever we talk about circles and spheres.  But
does it have a deeper significance?  Enter Buffon's
Needle ...

Take a sheet of paper marked with parallel lines
that are spaced a certain distance apart. Then take
a needle whose length must be less than the line
spacing.  Drop the needle onto the paper from a
reasonable height and record whether it lands (a)
not touching a line or (b) touching or crossing
a line.  Repeat this 100 times and record the
results.

Buffon used the results from his experiment with
a needle to estimate the value of π. He worked
out this formula:

π = 2L/x.p

Where,

L is the length of the needle, x is the line
spacing and p is the proportion of needles
crossing a line (case (b)).

So now we get the notion that π is not just
about circles, but has also has a statistical
significance.  In fact π shows up a lot in
statistics.  It is a key part of the equation
for the Normal Distribution.  So, π does indeed
have a deeper significance.