- Related Topics:
- Pi in nature
- pi theorem
- What Is Pi Day?
- circle
- Pi Day
pi
What is pi?
What is the value of pi?
What is the symbol for pi?
What are the uses of pi?
pi (π), in mathematics, the ratio of the circumference of a circle to its diameter. The Greek letter π was first used by British mathematician William Jones in 1706 to represent the ratio and was later popularized by Swiss mathematician Leonhard Euler. Because pi is irrational (not equal to the ratio of any two whole numbers), its digits do not repeat, and an approximation such as 3.14 or 22/7 is often used for everyday calculations. To 39 decimal places, pi is 3.141592653589793238462643383279502884197.
History
Early approximations
The Bible (1 Kings 7:23) describes the molten sea (a metal basin) of Solomon’s Temple as 10 cubits across and 30 cubits in circumference—an implied pi of 3.
The oldest surviving evidence of pi comes from Babylonian cuneiform tablets (c. 1900–1600 bce), which give a value for pi derived from a hexagon. They treated the ratio of an inscribed regular hexagon’s perimeter to the circle’s circumference as 24/25, giving the value of pi as 25/8, or about 3.125.
In Egypt, the Rhind papyrus (c. 1650 bce)—a scroll on arithmetic and geometry—uses the rule “subtract one-ninth of the diameter and square the remainder,” yielding 256/81, or about 3.16045.
" height="20" alt="HEW Logo">
Classical geometry
In the 3rd century bce, Greek mathematician and engineer Archimedes developed a geometric method to calculate pi. By comparing perimeters of polygons inscribed in and circumscribed around a circle (up to 96 sides), he proved 223/71 < π < 22/7, or an average value of about 3.1418. Archimedes also proved that the ratio of the area of a circle to the square of its radius is pi.
In the 2nd century ce, astronomer Claudius Ptolemy used trigonometric chord tables in his astronomical treatise Almagest to obtain a value of pi of about 3.1416.
Advances in Asia
In China, Liu Hui (3rd century ce) refined polygonal methods to calculate a value for pi of about 3.14159. Two centuries later, mathematician and astronomer Zu Chongzhi (429–500 ce) achieved a record level of accuracy with the fraction 355/113 and proved that pi lies between 3.1415926 and 3.1415927.
In India, Aryabhata (born 476 ce) used pi as 62,832/20,000 (about 3.1416) in the “Ganita” (“Mathematics”) section of his treatise Aryabhatiya. In Central Asia, astronomer Jamshid al-Kāshī (c. 1380–1429) calculated 2π to 9 sexagesimal (base 60) places, equivalent to about 16 decimal digits, giving the most precise value of pi then known.
The trusted destination for professionals, college students, and lifelong learners.
SUBSCRIBE
Early modern analysis
During the 16th and 17th centuries, European mathematicians developed new ways to calculate pi. French mathematician François Viète was the first to express pi as an endless product of numbers, introducing an approach that linked geometric ideas with algebraic calculation. In 1655–56, English mathematician John Wallis introduced an infinite product formula for pi, representing the constant as an unending sequence of ratios built from whole numbers.
In the early 1670s, James Gregory in Scotland and Gottfried Wilhelm Leibniz in Germany each presented a series for pi divided by 4 (1 – 1/3 + 1/5 – 1/7 + …). In 1706, John Machin of England used a faster arctangent formula to compute pi by hand to 100 decimal places.
Mathematical proofs and expanding precision
In 1768, Swiss mathematician Johann Heinrich Lambert proved that pi is irrational; it cannot be expressed as a ratio of whole numbers. German mathematician Ferdinand von Lindemann proved in 1882, that pi is transcendental, meaning it cannot be the root of any algebraic equation.
In 1873, English mathematician William Shanks computed 707 digits by hand, though later analysis showed errors beyond the 527th place. His results went unchallenged for 75 years, until American D.F. Ferguson, working with a mechanical desk calculator in 1946, identified the mistake and recomputed pi correctly to 808 digits.
In 1914, Indian mathematician Srinivasa Ramanujan presented 17 formulas for calculating pi, including a series for its reciprocal. Each term in the series produced several correct digits of pi, making his methods far more efficient than earlier approaches. His work later inspired the Ramanujan-Sato series and provided the mathematical foundation for modern computer algorithms that have computed pi to trillions of digits.
Pi Day, observed on March 14 (3/14), honors the mathematical constant π and its significance across science and engineering. The date reflects the first three digits of pi, 3.14, and coincides with Albert Einstein’s birthday. The tradition began in 1988, when physicist Larry Shaw organized the first celebration at the Exploratorium, a San Francisco science museum.
Computational era
In 1949, the Electronic Numerical Integrator and Computer (ENIAC) became the first electronic machine used to calculate pi, producing 2,037 digits in about 70 hours. As computers have become more powerful, they have calculated ever more digits of pi. In 2025, a record 314 trillion digits were calculated.