Pi Day and Archimedes

How Pi Day started

In 1988 Larry Shaw, a physicist who worked at the Exploratorium, a hands-on science museum in San Francisco, organised a small celebration for staff on 14 March: they walked in a circle and ate fruit pies. It became an annual event at the museum, where Shaw was nicknamed the “Prince of Pi”, and spread to schools, universities and maths clubs through the 1990s and the early internet. Shaw died in 2017.

On 12 March 2009 the US House of Representatives passed House Resolution 224, which supported the designation of a Pi Day and encouraged schools to use it to teach mathematics. It is a non-binding resolution, so Pi Day is not a federal holiday.

In November 2019 the General Conference of UNESCO proclaimed 14 March the International Day of Mathematics, at the proposal of the International Mathematical Union. The first was celebrated on 14 March 2020. Each year has a theme and an online map of events worldwide, which makes it the official, global version of Pi Day, including in countries that write the date as 14/3.

A few other dates sometimes come along: 22 July is “Pi Approximation Day” (22/7, a common fraction for pi), and fans of tau, which is two times pi, mark 28 June. 14 March is also the birthday of Albert Einstein (1879).

What pi is

Pi (π) is the ratio of any circle's circumference to its diameter. The ratio is the same for every circle, about 3.14159. It is irrational, which means its decimal digits never end and never fall into a repeating pattern (Johann Lambert proved this in 1761), and transcendental, which means it is not the solution of any polynomial equation with whole-number coefficients (Ferdinand von Lindemann, 1882). That second fact settled an ancient puzzle: you cannot “square the circle” with a ruler and compass.

The Greek letter π was first used for the ratio by the Welsh mathematician William Jones in 1706, and became standard after Leonhard Euler adopted it in the 1730s and 1740s.

How Archimedes trapped pi between two fractions

Archimedes of Syracuse (about 287 to 212 BC) wrote a short work called Measurement of a Circle. It survives in a damaged form, but it contains the first mathematical proof of how big pi is. Its third proposition states:

His method was to squeeze the circle between two polygons. Draw a regular hexagon inside a circle and another outside it. The circle's circumference must be longer than the inside hexagon's perimeter and shorter than the outside one's. With six sides the gap is wide (pi is between 3 and about 3.46). So he doubled the number of sides, to 12, then 24, 48 and finally 96, each time calculating the new perimeters by geometry. With no decimal numbers and no algebraic notation, he had to work out square roots by hand as fractions, and he always rounded in the safe direction so the bounds stayed true.

With 96-sided polygons he got 3 10/71 (about 3.1408) as a lower limit and 3 1/7 (about 3.1429) as an upper limit. The true value, 3.14159, sits between them. The upper bound, 22/7, is still the fraction most students learn.

The same treatise proves another result in its first proposition: the area of a circle equals the area of a right-angled triangle whose two shorter sides are the radius and the circumference. In modern terms, that is area = πr². Archimedes was proudest of a related discovery, that a sphere has two-thirds the volume of the cylinder that just contains it. According to Plutarch and Cicero, a sphere inside a cylinder was carved on his tomb, and Cicero says he found the overgrown tomb in Syracuse in 75 BC.

Pi after Archimedes: a short timeline

For the Indian part of the story see Aryabhata and our article on National Mathematics Day and Ramanujan. Ramanujan's 1914 series were so efficient that the programmer Bill Gosper used one in 1985 to compute more than 17 million digits.

Pi Day activities that teach something

• Measure it yourself: students wrap string round tins, plates and wheels, measure the diameters, and divide. Pool the class results and see how close the average gets to 3.14.
• Do Archimedes' first step: draw a circle of radius 5 cm with a hexagon inside and a square outside. Measure both perimeters and divide by the diameter to get a lower and an upper bound. Then try an octagon.
• Spreadsheet polygons: older students can use the doubling formula for polygon perimeters in a spreadsheet and watch the bounds close in at 6, 12, 24, 48 and 96 sides, as Archimedes did by hand.
• Buffon's needle: drop toothpicks on lined paper (lines one toothpick-length apart) and count crossings. Twice the number of drops divided by the number of crossings approaches pi.
• Fraction race: which is closer to pi, 22/7 or 355/113? Students compute both and the error of each.
• Ask Archimedes: students write questions about his proof and his life in Syracuse and practise them by voice with an AI recreation of Archimedes on Yuga, built from his surviving works and ancient accounts. It is not Archimedes, and stories about him, like the bath and “Eureka”, are legends, so check claims against the texts.

For the bath story, see Did Archimedes shout “Eureka”?. Our questions to ask Archimedes list has more ideas for an interview project.