The most well-known fact about pi—normally rounded to 3.14159—is that it represents the ratio of the circumference of a circle to its diameter. Pi is also an irrational number, so it is incapable of being written as a simple fraction. Therefore, pi is an infinitely long, non-repeating decimal, making it one of the most interesting and mysterious numbers known to man.

## 10. First Calculation

The first calculation of pi is believed to have been obtained by Archimedes of Syracuse around 220 BC. Archimedes derived the formula A = pi r^{2} by approximating the area of a circle based on the area of a regular polygon inscribed within the circle, and the area of a polygon within which the circle was circumscribed. The two polygons therefore provided the upper and lower bounds for the area of a circle—allowing Archimedes to approximate that the missing piece of the puzzle (pi) lay somewhere between 3 1/7 and 3 10/71.

The prominent Chinese mathematician and astronomer Zu Chongzi (429–501) later calculated pi to be 355/113, although exactly how he was able to reach this incredibly precise measurement remains a mystery, since there are no records of his work.

## 9. A Circle’s True Area Is Unknowable

Johann Heinrich Lambert in the 18th century, proved that pi is irrational—it cannot be expressed as an integer-based fraction. Rational numbers can always be written as a fraction, in which both the numerator and the denominator are whole numbers. Although it might be tempting to view pi as a simple ratio of circumference/diameter (pi=C/D), it will always be the case that if the diameter is an integer, the circumference is not an integer, and vice versa.

The irrationality of pi means we can never truly know the circumference (and subsequently the area) of a circle. This frustrating yet seemingly unavoidable fact has led some mathematicians to insist that it is more accurate to think of a circle as having an infinite number of tiny corners, instead of thinking of a circle as being “smooth.”

## 8. Buffon’s Needle

First brought to the attention of geometricians and mathematicians in 1777, Buffon’s needle is one of the oldest and most intriguing problems in the field of geometrical probability. Here’s how it works.

If you were to drop a needle of one unit length onto a sheet of paper with lines separated by the same single unit length, the probability that the needle crosses one of the lines on the page is directly related to the value of pi.

There are two variables involved in the needle drop: 1) the angle at which the needle falls, and 2) the distance from the center of the needle to the closest line. The angle can vary from 0 to 180 degrees and is measured against a line parallel to the lines on the paper.

It turns out the probability that the needle lands so that it cuts a line is exactly 2/pi, or roughly 64 percent. This means that pi could be theoretically calculated using this technique if one had enough patience to sit through enough trials, even though the experiment seems to have nothing to do with circles, or even rounded edges for that matter.

This may be a bit difficult to envision, so experiment with the phenomenon yourself here.