What Is Pi? Meaning, Value, Uses and History

Pi turns up wherever something is round, curved or repeating, from NASA mission planning to the bell curve¹⁵. Below: the value and definition first, then the formulas, the digits anyone actually needs, the uses, and a history that runs from a temple casting to a computation of 100 trillion digits³⁷.

Pi at a glance

What is pi?

Pi is the circumference of a circle divided by the circle’s diameter, and it is the same for any circle, no matter how big or small¹. The circumference is the distance around a circle, and the diameter is a straight line across it through the center¹. Put another way, pi counts how many diameters fit around the rim¹.

Three diameters fit around the rim with a little left over. With pi near 3.14, the leftover is about 0.14 of a diameter (our arithmetic)¹.

Worked example (our arithmetic): the same ratio for two pizzas, using C = π × d, A = π × r × r with the full value of pi¹⁹

The rim grows in step with the diameter, and the area grows with the square of the radius⁹. That is why a 16 inch pizza is not 1.33 times the small one but 1.78 times as much to eat (our arithmetic)⁹.

Pi formulas

Radians are the reason pi shows up in trigonometry and calculus: dividing a circle into 360 degrees is an arbitrary choice, while the radian is built from the circle itself⁶. The same constant sits inside the formula for the area under a normal curve, a calculus result⁵. For the calculus ideas behind infinite digit-by-digit calculations, see what a limit is.

Try a circle with a rounded pi

Which value of pi is good enough?

Usually the approximation 3.14 is precise enough, even for NASA scientists and engineers¹. For its highest accuracy calculations, interplanetary navigation, the JPL education team uses 3.141592653589793². Rounding pi to 3.14 changes a result by a small, predictable fraction, and three examples from JPL show how few digits the real world needs.

Worked example (our arithmetic): what rounding to 3.14 costs on Earth’s circumference¹²

Twenty kilometres is too much for a navigation fix and nothing for a pizza. That gap is why precision depends on the job, and why world record holders may have memorized more than 70,000 digits of pi while a JPL engineer explains you only need a tiny fraction for most calculations².

What is the actual purpose of pi?

Pi is fundamental for calculating anything that involves circles, curves or spheres, and it is used in geometry, physics, engineering and even computer science¹. Its purpose is to link a straight measurement to a round one.

Why pi never ends

Pi is an irrational number, so it goes on forever and never repeats its sequence of numbers¹. An irrational number is one that cannot be expressed as a simple fraction¹. Because the decimal expansion for pi goes on forever with no discernible pattern, that is one of the hallmarks of an irrational number⁵. It was suspected for many years that pi was irrational, and Johann Lambert proved it in 1761⁵³. Wikipedia dates his proof to 1768⁸, and this page keeps the 1761 date that both the college and the history source give.

Pi is also transcendental, as the MacTutor history and others record. Ferdinand von Lindemann proved this in 1882⁸, and it means pi is not the solution of any polynomial equation with integer coefficients³. A square equal in area to a given circle cannot be built with a ruler and compass, which ended the ancient quest to square the circle³.

The digits of pi

The first 100 decimals of pi, from a university reference table of 2000 places⁴, are:

3.14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679⁴

Those digits have so far passed all statistical tests for randomness³, and detailed analyses found them consistent with normality⁸. Whether pi is normal, meaning every block of digits shows up equally often, has not been proven or disproven⁸. In the first 200 decimals, the pair 67 appears exactly once, at the 98th decimal⁴. Because the digits behave randomly, a short pair like that has no special meaning.

History of pi

From Solomon’s casting to Archimedes

The temple verse reads as a round basin ten cubits across with a line of thirty cubits around it, and that gives pi equal to 3³. The earliest values of pi, including this one, were almost certainly found by measurement³. The ratio itself has been known for so long that its discovery is untraceable³.

Archimedes made the first theoretical calculation, drawing polygons inside and outside the circle and doubling the number of sides from a hexagon to 96 sides⁸³. He bracketed pi between 3.1408 and 3.1429⁸. The achievement, in the words of one history, is not that he stopped at 96 sides but that he went so far³.

Worked example (our arithmetic): the Archimedes bracket and its midpoint⁸

Records and a few false starts

Computing digits became a contest. Shanks spent years on 707 places and published them in 1873, but only 527 were correct, so 180 were wrong (our arithmetic)³. In 1945 Ferguson found that the error began at the 528th place³. In 1949 a computer reached 2,037 digits on ENIAC in 70 hours⁸. Google Cloud announced a 100 trillion digit record in 2022⁸, after a run of 157 days, 23 hours and 31 minutes⁷.

Worked example (our arithmetic): the speed of the 100 trillion digit record⁷

In 1897 the Indiana House of Representatives unanimously passed a bill introducing a new mathematical truth about circles, and the Indiana Senate postponed it indefinitely³.

Pi Day

Pi Day is March 14, because March is the third month and 3/14 reads as 3.14, as a Randolph-Macon College mathematician explains⁵. A physicist in San Francisco, Larry Shaw, held the first one on March 14, 1988⁵. In 2009 the United States Congress declared March 14 national Pi Day⁵. NASA also runs a yearly challenge of illustrated problems where students use pi the way its scientists and engineers do².

Common mix-ups

• Pi equals a simple fraction. It does not: pi cannot be expressed as a simple fraction¹, and any fraction used for it is a rounding⁸.
• Pi is 3.14. The rounding is usually precise enough¹ but 3.14 is off by about 0.051 percent from the real value (our arithmetic)¹.
• Pi has a hidden pattern. The digits have no discernible pattern and pass statistical randomness tests⁵³.
• More digits make a better circle. For the known universe, 37 decimal places already reach the width of an atom².

For other places where one fixed number drives a daily experience, read why seasons happen, where a steady tilt of the axis drives the year, why power lines use high voltage, and how fast coffee cools. See also the world section.