Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century. Further progress was not made until the 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega). The record of manual approximation of π is held by William Shanks, who calculated 527 decimals correctly in 1853. Since the middle of the 20th century, the approximation of π has been the task of electronic digital computers (for a comprehensive account, see chronology of computation of pi). On December 11, 2025, the current record was established by StorageReview with Alexander Yee's y-cruncher with 314 trillion (3.14×1014) digits.
Early history The best known approximations to π dating to before the Common Era were accurate to two decimal places; this was improved upon in Chinese mathematics in particular by the mid-first millennium, to an accuracy of seven decimal places. After this, no further progress was made until the late medieval period. Some Egyptologists have claimed that the ancient Egyptians used an approximation of π as 22⁄7 = 3.142857 (about 0.04% too high) from as early as the Old Kingdom (c. 2700–2200 BC). This claim has been met with skepticism. Babylonian mathematics usually approximated π to 3, sufficient for the architectural projects of the time (notably also reflected in the description of Solomon's Temple in the Hebrew Bible). The Babylonians were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near Susa in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of π as 25⁄8 = 3.125, about 0.528% below the exact value. At about the same time, the Egyptian Rhind Mathematical Papyrus (dated to the Second Intermediate Period, c. 1600 BCE, although stated to be a copy of an older, Middle Kingdom text) implies an approximation of π as 256⁄81 ≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle via approximation with the octagon. Astronomical calculations in the Shatapatha Brahmana (c. 6th century BCE) use a fractional approximation of 339⁄108 ≈ 3.139. The Mahabharata (500 BCE – 300 CE) offers an approximation of 3, in the ratios offered in Bhishma Parva verses: 6.12.40–45.
... The Moon is handed down by memory to be eleven thousand yojanas in diameter. Its peripheral circle happens to be thirty three thousand yojanas when calculated. ... The Sun is eight thousand yojanas and another two thousand yojanas in diameter. From that its peripheral circle comes to be equal to thirty thousand yojanas.
... In the 3rd century BCE, Archimedes proved the sharp inequalities 223⁄71 < π < 22⁄7 , by means of regular 96-gons (accuracies of 2·10−4 and 4·10−4, respectively). In the 2nd century CE, Ptolemy used the value 377⁄120, the first known approximation accurate to three decimal places (accuracy 2·10−5). It is equal to 3 + 8 / 60 + 30 / 60 2 , {\displaystyle 3+8/60+30/60^{2},} which is accurate to two sexagesimal digits. The Chinese mathematician Liu Hui in 263 CE computed π to between 3.141024 and 3.142708 by inscribing a 96-gon and 192-gon; the average of these two values is 3.141866 (accuracy 9·10−5). He also suggested that 3.14 was a good enough approximation for practical purposes. He has also frequently been credited with a later and more accurate result, π ≈ 3927⁄1250 = 3.1416 (accuracy 2·10−6), although some scholars instead believe that this is due to the later (5th-century) Chinese mathematician Zu Chongzhi. Zu Chongzhi is known to have computed π to be between 3.1415926 and 3.1415927, which was correct to seven decimal places. He also gave two other approximations of π: π ≈ 22⁄7 and π ≈ 355⁄113, which are not as accurate as his decimal result. The latter fraction is the best possible rational approximation of π using fewer than five decimal digits in the numerator and denominator. Zu Chongzhi's results surpass the accuracy reached in Hellenistic mathematics, and would remain without improvement for close to a millennium. In Gupta-era India (6th century), mathematician Aryabhata, in his astronomical treatise Āryabhaṭīya stated:
Add 4 to 100, multiply by 8 and add to 62,000. This is 'approximately' the circumference of a circle whose diameter is 20,000. Approximating π to four decimal places: π ≈ 62832⁄20000 = 3.1416, Aryabhata stated that his result "approximately" (āsanna "approaching") gave the circumference of a circle. His 15th-century commentator Nilakantha Somayaji (Kerala school of astronomy and mathematics) has argued that the word means not only that this is an approximation, but that the value is incommensurable (irrational).
Middle Ages Further progress was not made for nearly a millennium, until the 14th century, when Indian mathematician and astronomer Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics, found the Maclaurin series for arctangent, and then two infinite series for π. One of them is now known as the Madhava–Leibniz series, based on π = 4 arctan ( 1 ) : {\displaystyle \pi =4\arctan(1):}
π = 4 ( 1 − 1 3 + 1 5 − 1 7 + ⋯ ) {\displaystyle \pi =4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots \right)}
The other was based on π = 6 arctan ( 1 / 3 ) : {\displaystyle \pi =6\arctan(1/{\sqrt {3}}):}
π = 12 ∑ k = 0 ∞ ( − 3 ) − k 2 k + 1 = 12 ∑ k = 0 ∞ ( − 1 3 ) k 2 k + 1 = 12 ( 1 − 1 3 ⋅ 3 + 1 5 ⋅ 3 2 − 1 7 ⋅ 3 3 + ⋯ ) {\displaystyle \pi ={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-3)^{-k}}{2k+1}}={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-{\frac {1}{3}})^{k}}{2k+1}}={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)}
He used the first 21 terms to compute an approximation of π correct to 11 decimal places as 3.14159265359. He also improved the formula based on arctan(1) by including a correction:
π / 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ − ( − 1 ) n 2 n − 1 ± n 2 + 1 4 n 3 + 5 n {\displaystyle \pi /4\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots -{\frac {(-1)^{n}}{2n-1}}\pm {\frac {n^{2}+1}{4n^{3}+5n}}}
It is not known how he came up with this correction. Using this he found an approximation of π to 13 decimal places of accuracy when n = 75. Indian mathematician Bhaskara II (12th century) used regular polygons with up to 384 sides to obtain another approximation of π, calculating it as 3.141666. Jamshīd al-Kāshī (Kāshānī) (15th century), a Persian astronomer and mathematician, correctly computed the fractional part of 2π to 9 sexagesimal digits in 1424, and translated this into 16 decimal digits after the decimal point:
2 π ≈ 6.2831853071795864 , {\displaystyle 2\pi \approx 6.2831853071795864,}
which gives 16 correct digits for π after the decimal point:
π ≈ 3.1415926535897932 {\displaystyle \pi \approx 3.1415926535897932}
He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 228 sides.
16th to 19th centuries In the second half of the 16th century, the French mathematician François Viète discovered an infinite product that converged on π known as Viète's formula. The German-Dutch mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimal places of π with a 262-gon. He was so proud of this accomplishment that he had the digits inscribed on his tombstone. In Cyclometricus (1621), Willebrord Snellius demonstrated that the perimeter of the inscribed polygon converges on the circumference twice as fast as does the perimeter of the corresponding circumscribed polygon. This was proved by Christiaan Huygens in 1654. Snellius was able to obtain seven digits of π from a 96-sided polygon. In 1656, John Wallis published the Wallis product:
π 2 = ∏ n = 1 ∞ 4 n 2 4 n 2 − 1 = ∏ n = 1 ∞ ( 2 n 2 n − 1 ⋅ 2 n 2 n + 1 ) = ( 2 1 ⋅ 2 3 ) ⋅ ( 4 3 ⋅ 4 5 ) ⋅ ( 6 5 ⋅ 6 7 ) ⋅ ( 8 7 ⋅ 8 9 ) ⋅ ⋯ {\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdot \;\cdots }
In 1706, John Machin used Gregory's series (the Taylor series for arctangent) and the identity 1 4 π = 4 arccot 5 − arccot 239 {\textstyle {\tfrac {1}{4}}\pi =4\operatorname {arccot} 5-\operatorname {arccot} 239} to calculate 100 digits of π (see § Machin-like formula below). In 1719, Thomas de Lagny used a similar identity to calculate 127 digits (of which 112 were correct). In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula to calculate the first 140 digits, of which the first 126 were correct. In 1841, William Rutherford calculated 208 digits, of which the first 152 were correct. The magnitude of such precision (152 decimal places) can be put into context by the fact that the circumference of the largest known object, the observable universe, can be calculated from its diameter (93 billion light-years) to a precision of less than one Planck length (at 1.6162×10−35 meters, the shortest unit of length expected to be directly measurable) using π expressed to just 62 decimal places. The English amateur mathematician William Shanks calculated π to 530 decimal places in January 1853, of which the first 527 were correct (the last few likely being incorrect due to round-off errors). He subsequently expanded his calculation to 607 decimal places in April 1853, but an error introduced right at the 530th decimal place rendered the rest of his calculation erroneous; due to the nature of Machin's formula, the error propagated back to the 528th decimal place, leaving only the first 527 digits correct once again. Twenty years later, Shanks expanded his calculation to 707 decimal places in April 1873. Due to this being an expansion of his previous calculation, most of the new digits were incorrect as well. Shanks was said to have calculated new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of π until the advent of the electronic digital computer three-quarters of a century later.
20th and 21st centuries
In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of π, including
1 π = 2 2 9801 ∑ k = 0 ∞ ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) 4 396 4 k {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}}
which computes a further eight decimal places of π with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate π. Evaluating the first term alone yields a value correct to seven decimal places:
π ≈ 9801 2206 2 ≈ 3.14159273 {\displaystyle \pi \approx {\frac {9801}{2206{\sqrt {2}}}}\approx 3.14159273}
See Ramanujan–Sato series. From the mid-20th century onwards, all improvements in calculation of π have been done with the help of calculators or computers. In 1944−45, D. F. Ferguson, with the aid of a mechanical desk calculator, found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect. In the early years of the computer, an expansion of π to 100000 decimal places was computed by Maryland mathematician Daniel Shanks (no relation to the aforementioned William Shanks) and his team at the United States Naval Research Laboratory in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating the digits of π. For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of π were published in 1962. The authors outlined what would be needed to calculate π to 1 million decimal places and concluded that the task was beyond that day's technology, but would be possible in five to seven years. In 1989, the Chudnovsky brothers computed π to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of π:
1 π = 12 ∑ k = 0 ∞ ( − 1 ) k ( 6 k ) ! ( 13591409 + 545140134 k ) ( 3 k ) ! ( k ! ) 3 640320 3 k + 3 / 2 . {\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}.}
Records since then have all been accomplished using the Chudnovsky algorithm. In 1999, Yasumasa Kanada and his team at the University of Tokyo computed π to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of π. In November 2002, Yasumasa Kanada and a team of 9 others used the Hitachi SR8000, a 64-node supercomputer with 1 terabyte of main memory, to calculate π to roughly 1.24 trillion digits in around 600 hours (25 days).
Recent records In August 2009, a Japanese supercomputer called the T2K Open Supercomputer more than doubled the previous record by calculating π to roughly 2.6 trillion digits in approximately 73 hours and 36 minutes. In December 2009, Fabrice Bellard used a home computer to compute 2.7 trillion decimal digits of π. Calculations were performed in base 2 (binary), then the result was converted to base 10 (decimal). The calculation, conversion, and verification steps took a total of 131 days. In August 2010, Shigeru Kondo used Alexander Yee's y-cruncher to calculate 5 trillion digits of π. This was the world record for any type of calculation, but significantly it was performed on a home computer built by Kondo. The calculation was done between 4 May and 3 August, with the primary and secondary verifications taking 64 and 66 hours respectively. In October 2011, Shigeru Kondo broke his own record by computing ten trillion (1013) and fifty digits using the same method but with better hardware. In December 2013, Kondo broke his own record for a second time when he computed 12.1 trillion digits of π. In October 2014, Sandon Van Ness, going by the pseudonym "houkouonchi" used y-cruncher to calculate 13.3 trillion digits of π. In November 2016, Peter Trueb and his sponsors computed on y-cruncher and fully verified 22.4 trillion digits of π (22,459,157,718,361 (πe × 1012)). The computation took (with three interruptions) 105 days to complete, the limitation of further expansion being primarily storage space. In March 2019, Emma Haruka Iwao, an employee at Google, computed 31.4 (approximately 10π) trillion digits of pi using y-cruncher and Google Cloud machines. This took 121 days to complete. In January 2020, Timothy Mullican announced the computation of 50 trillion digits over 303 days. On 14 August 2021, a team (DAViS) at the University of Applied Sciences of the Grisons announced completion of the computation of π to 62.8 (approximately 20π) trillion digits. On 8 June 2022, Emma Haruka Iwao announced on the Google Cloud Blog the computation of 100 trillion (1014) digits of π over 158 days using Alexander Yee's y-cruncher. On 14 March 2024, Jordan Ranous, Kevin O’Brien and Brian Beeler computed π to 105 trillion digits, also using y-cruncher. On 28 June 2024, the StorageReview Team computed π to 202 trillion digits, also using y-cruncher. On 2 April 2025, Linus Media Group and Kioxia computed π to 300 trillion digits, also using y-cruncher. On 11 December 2025, the record returned to the StorageReview Team, after they computed π to 314 trillion digits, again using y-cruncher.
Practical approximations Depending on the purpose of a calculation, π can be approximated by using fractions for ease of calculation. The most notable such approximations are 22⁄7 (relative error of about 4·10−4) and 355⁄113 (relative error of about 8·10−8). In Chinese mathematics, the fractions 22/7 and 355/113 are known as Yuelü (约率; yuēlǜ; 'approximate ratio') and Milü (密率; mìlǜ; 'close ratio').
Non-mathematical "definitions" of π Of some notability are legal or historical texts purportedly "defining π" to have some rational value, such as the "Indiana Pi Bill" of 1897, which stated "the ratio of the diameter and circumference is as five-fourths to four" (which would imply "π = 3.2") and a passage in the Hebrew Bible that implies that π = 3.
Indiana bill
The so-called "Indiana Pi Bill" from 1897 has often been characterized as an attempt to "legislate the value of Pi". Rather, the bill dealt with a purported solution to the problem of geometrically "squaring the circle". The bill was nearly passed by the Indiana General Assembly in the U.S., and has been claimed to imply a number of different values for π, although the closest it comes to explicitly asserting one is the wording "the ratio of the diameter and circumference is as five-fourths to four", which would make π = 16⁄5 = 3.2, a discrepancy of nearly 2 percent. A mathematics professor who happened to be present the day the bill was brought up for consideration in the Senate, after it had passed in the House, helped to stop the passage of the bill on its second reading, after which the assembly thoroughly ridiculed it before postponing it indefinitely.
Imputed biblical value
It is sometimes claimed that the Hebrew Bible implies that "π equals three", based on a passage in 1 Kings 7:23 and 2 Chronicles 4:2 giving measurements for the round basin located in front of the Temple in Jerusalem as having a diameter of 10 cubits and a circumference of 30 cubits. The issue is discussed in the Talmud and in Rabbinic literature. Among the many explanations and comments are these:
Rabbi Nehemiah explained this in his Mishnat ha-Middot (the earliest known Hebrew text on geometry, ca. 150 CE) by saying that the diameter was measured from the outside rim while the circumference was measured along the inner rim. This interpretation implies a brim about 0.225 cubit (or, assuming an 18-inch "cubit", some 4 inches), or one and a third "handbreadths," thick (cf. NRSV and NRSV). Maimonides states (ca. 1168 CE) that π can only be known approximately, so the value 3 was given as accurate enough for religious purposes. This is taken by some as the earliest assertion that π is irrational. There is still some debate on this passage in biblical scholarship. Many reconstructions of the basin show a wider brim (or flared lip) extending outward from the bowl itself by several inches to match the description given in NRSV In the succeeding verses, the rim is described as "a handbreadth thick; and the brim thereof was wrought like the brim of a cup, like the flower of a lily: it received and held three thousand baths" NRSV, which suggests a shape that can be encompassed with a string shorter than the total length of the brim, e.g., a Lilium flower or a Teacup.
Development of efficient formulae
Polygon approximation to a circle Archimedes, in his Measurement of a Circle, created the first algorithm for the calculation of π based on the idea that the perimeter of any (convex) polygon inscribed in a circle is less than the circumference of the circle, which, in turn, is less than the perimeter of any circumscribed polygon. He started with inscribed and circumscribed regular hexagons, whose perimeters are readily determined. He then shows how to calculate the perimeters of regular polygons of twice as many sides that are inscribed and circumscribed about the same circle. This is a recursive procedure which would be described today as follows: Let pk and Pk denote the perimeters of regular polygons of k sides that are inscribed and circumscribed about the same circle, respectively. Then,
P 2 n = 2 p n P n p n + P n , p 2 n = p n P 2 n . {\displaystyle P_{2n}={\frac {2p_{n}P_{n}}{p_{n}+P_{n}}},\quad \quad p_{2n}={\sqrt {p_{n}P_{2n}}}.}
Archimedes uses this to successively compute P12, p12, P24, p24, P48, p48, P96 and p96. Using these last values he obtains
3 10 71 < π < 3 1 7 . {\displaystyle 3{\frac {10}{71}}<\pi <3{\frac {1}{7}}.}
It is not known why Archimedes stopped at a 96-sided polygon; it only takes patience to extend the computations. Heron reports in his Metrica (about 60 CE) that Archimedes continued the computation in a now lost book, but then attributes an incorrect value to him. Archimedes uses no trigonometry in this computation and the difficulty in applying the method lies in obtaining good approximations for the square roots that are involved. Trigonometry, in the form of a table of chord lengths in a circle, was probably used by Claudius Ptolemy of Alexandria to obtain the value of π given in the Almagest (circa 150 CE). Advances in the approximation of π (when the methods are known) were made by increasing the number of sides of the polygons used in the computation. A trigonometric improvement by Willebrord Snell (1621) obtains better bounds from a pair of bounds obtained from the polygon method. Thus, more accurate results were obtained from polygons with fewer sides. Viète's formula, published by François Viète in 1593, was derived by Viète using a closely related polygonal method, but with areas rather than perimeters of polygons whose numbers of sides are powers of two. The last major attempt to compute π by this method was carried out by Grienberger in 1630 who calculated 39 decimal places of π using Snell's refinement.
Machin-like formula
For fast calculations, one may use formulae such as Machin's:
π 4 = 4 arctan 1 5 − arctan 1 239 {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}
together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, producing:
( 5 + i ) 4 ⋅ ( 239 − i ) = 2 2 ⋅ 13 4 ( 1 + i ) . {\displaystyle (5+i)^{4}\cdot (239-i)=2^{2}\cdot 13^{4}(1+i).}
((x),(y) = {239, 132} is a solution to the Pell equation x2 − 2y2 = −1.) Formulae of this kind are known as Machin-like formulae. Machin's particular formula was used well into the computer era for calculating record numbers of digits of π, but more recently other similar formulae have been used as well. For instance, Shanks and his team used the following Machin-like formula in 1961 to compute the first 100,000 digits of π:
π 4 = 6 arctan 1 8 + 2 arctan 1 57 + arctan 1 239 {\displaystyle {\frac {\pi }{4}}=6\arctan {\frac {1}{8}}+2\arctan {\frac {1}{57}}+\arctan {\frac {1}{239}}}
and they used another Machin-like formula,
π 4 = 12 arctan 1 18 + 8 arctan 1 57 − 5 arctan 1 239 {\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{18}}+8\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}}
as a check. The record as of December 2002 by Yasumasa Kanada of Tokyo University stood at 1,241,100,000,000 digits. The following Machin-like formulae were used for this:
π 4 = 12 arctan 1 49 + 32 arctan 1 57 − 5 arctan 1 239 + 12 arctan 1 110443 {\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}}
K. Takano (1982).
π 4 = 44 arctan 1 57 + 7 arctan 1 239 − 12 arctan 1 682 + 24 arctan 1 12943 {\displaystyle {\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}}
F. C. M. Størmer (1896).
Other classical formulae Other formulae that have been used to compute estimates of π include: Liu Hui (see also Viète's formula):
π ≈ 768 2 − 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 1 ≈ 3.14159046323. {\displaystyle {\begin{aligned}\pi &\approx 768{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+1}}}}}}}}}}}}}}}}}}\\&\approx 3.14159046323.\end{aligned}}}
Madhava:
π = 12 ∑ k = 0 ∞ ( − 3 ) − k 2 k + 1 = 12 ∑ k = 0 ∞ ( − 1 3 ) k 2 k + 1