Inspired by Colin Wright, here's a Python list comprehension working out the top 10 date approximations:
from math import pi
print [i[1] for i in sorted([
(abs(pi - (day + 1.) / (month + 1)), "{}/{}".format(day + 1, month + 1))
for month in range(12)
for day in range([31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31][month])
])[:10]]