Pi Formulas

Pi formulas take many forms. Among others, these include series, products, geometric constructions involving , limits, special values, and pi iterations.

is intimately related to the properties of circles and spheres. For a circle of radius , the circumference and area are given by

Similarly, for a sphere of radius , the surface area and volume enclosed are

One of the earliest rapidly convergent formulas is Newton's pi series, derived in 1666 from the binomial series and the area of a circular segment,

(Wells 1986, p. 50; Borwein et al. 1989; Borwein and Bailey 2003, pp. 105-106; Veritasium 2021).

An exact formula for in terms of the inverse tangents of unit fractions is Machin's formula

There are three other Machin-like formulas, as well as thousands of other similar formulas having more terms.

Gregory and Leibniz found

(Wells 1986, p. 50), which is known as the Gregory series and may be obtained by plugging into the Leibniz series for . The absolute error after terms of the series for lies strictly between and . Multiplying the 300-term sum by 4 gives , with an absolute error of approximately 0.0033333. The series can be transformed to

where is the Riemann zeta function (Vardi 1991, pp. 157-158; Flajolet and Vardi 1996), so that the error after terms is asymptotic to .

A series attributed to Abraham Sharp (ca. 1717) is

(Smith 1953, p. 311). Additional simple series in which appears are

(Wells 1986, p. 53).

An application of Euler's convergence improvement transformation gives

(Beeler et al. 1972, Item 120).

This corresponds to plugging into the power series for the hypergeometric function ,

where denotes the double factorial. Despite the convergence improvement, series (19) converges at only one bit per term. At the cost of a square root, Gosper has noted that gives 2 bits per term,

and gives almost 3.39 bits per term,

where is the golden ratio.

Another series involving the golden ratio is

This was communicated by B. Cloitre (pers. comm., Dec. 13, 2007). Pardeshi (2022) gives an equivalent form by extracting the first term and shifting the remaining terms with denominator to denominator . For a short proof, the inverse tangent addition formula gives . Expanding the two inverse tangents in their absolutely convergent Taylor series and grouping terms by their exponents modulo 12 gives the result.

Gosper also obtained

A rapidly convergent series given by Hakimoglu-Brown (2024) is

where is the Pochhammer symbol. The summands are rational numbers, with an algebraic prefactor . The ratio of successive terms tends to , giving approximately decimal digits per term. An equivalent form posed by Sun (2024) is

The two forms agree term by term after the index shift and their respective prefactors are included.

Various limits also converge to , a simple example being

More interesting examples are given by

and

where is a Bernoulli number (Plouffe 2022). These formulas give rapidly converging approximations to ; certifying pi digits from an approximation requires an error bound that excludes crossing a digit-truncation boundary.

A spigot algorithm for is given by Rabinowitz and Wagon (1995; Borwein and Bailey 2003, pp. 141-142).

An explicit series representation giving another digit-extraction algorithm which produces digits of (or ) in base 16 was discovered by Bailey et al. (Bailey et al. 1997, Adamchik and Wagon 1997),

This formula, known as the BBP formula, was discovered using the PSLQ algorithm (Ferguson et al. 1999) and is equivalent to

There are BBP-type formulas for with alternating signs, including

Similarly, examples of BBP-type formulas for with bases that are powers of 2 are

Bellard (1997a) found the rapidly converging BBP-type formula

A related integral is

(Dalzell 1944, 1971; Le Lionnais 1983, p. 22; Borwein et al. 2004a, p. 3; Boros and Moll 2004, p. 125; Lucas 2005; Borwein et al. 2007, p. 14). This integral was known by K. Mahler in the mid-1960s and appears in an exam at the University of Sydney in November 1960 (Borwein et al. 2004a, p. 3). For the next convergent, Lucas (2009) obtained

The fourth convergent satisfies

(Lucas 2005; Bailey et al. 2007, p. 219). Lucas (2005, 2009) gives further such integrals.

Backhouse (1995) used the identity

for positive integers and , where , , and are rational constants to generate a number of formulas for . In particular, if , then (Lucas 2005).

A similar formula was subsequently discovered by Ferguson, leading to a two-dimensional lattice of such formulas which can be generated by these two formulas given by

for any complex value of (Adamchik and Wagon), giving the BBP formula as the special case .

An even more general identity due to Wagon is given by

(Borwein and Bailey 2003, p. 141). With principal branches on the left, the series converges and the identity holds for . Analytic continuation beyond this disk depends on branch choices, as illustrated above.

A perhaps even stranger general class of identities is given by

which holds for any positive integer , where is a Pochhammer symbol (B. Cloitre, pers. comm., Jan. 23, 2005). Even more amazingly, there is a closely analogous formula for the natural logarithm of 2.

Following the discovery of the base 16 digit BBP formula and related formulas, similar formulas in other bases were investigated. Borwein et al. (2004b) proved that, for an integer base that is not a perfect power, has no Machin-type BBP formula expressing it as a finite rational linear combination of with positive integers . This does not rule out other digit-extraction algorithms in such bases.

S. Plouffe has devised an algorithm to compute the th digit of in any base in steps.

A slew of additional identities due to Ramanujan, Catalan, and Newton are given by Castellanos (1988ab, pp. 86-88), including several involving sums of Fibonacci numbers. Ramanujan found

(Hardy 1923, 1924, 1999, p. 7).

Plouffe (2006) found the beautiful formula

An interesting infinite product formula due to Euler which relates and the th prime is

(Blatner 1997, p. 119), plotted above as a function of the number of terms in the product.

A method similar to Archimedes' can be used to estimate by starting with a regular polygon with sides and repeatedly doubling the number of sides. Let be half the central angle subtended by one side. Repeated use of the double-angle formulas gives

(Beckmann 1989, pp. 92-94).

Vieta (1593) was the first to give an exact expression for by taking in the above expression, giving

which leads to an infinite product of nested radicals,

(Wells 1986, p. 50; Beckmann 1989, p. 95). Rudio (1891) gave a rigorous proof of its convergence.

A related formula is given by

which can be written

where is defined using the iteration

with (J. Munkhammar, pers. comm., Apr. 27, 2000). The formula

is also closely related.

A pretty formula for is given by

where the numerator is a form of the Wallis formula for and the denominator is a telescoping sum with sum 1/2 since

(Sondow 1997).

A particular case of the Wallis formula gives

(Wells 1986, p. 50). This formula can also be written

where denotes a binomial coefficient and is the gamma function (Knopp 1990). Euler obtained

which follows from the special value of the Riemann zeta function . Similar formulas follow from for all positive integers .

An infinite sum due to Ramanujan is

(Borwein et al. 1989; Borwein and Bailey 2003, p. 109; Bailey et al. 2007, p. 44). Further sums are given in Ramanujan (1913-14),

and

(Beeler et al. 1972, Item 139; Borwein et al. 1989; Borwein and Bailey 2003, p. 108; Bailey et al. 2007, p. 44). Equation (80) is derived from a modular identity of order 58, although a first derivation was not presented prior to Borwein and Borwein (1987a). The initial-term approximations from the two series are, respectively,

The series provide about 6 and 8 decimal digits per term, respectively (Beeler et al. 1972, Item 139; Wells 1986, p. 54). Such series exist because of the rationality of various modular invariants.

A Chudnovsky-type family of series is

where is a positive integer satisfying , is a negative binary quadratic form discriminant, and . Here denotes the j-function evaluated at , and the family is taken with so that the series converges. Define

where denotes the Eisenstein series evaluated at (Borwein and Borwein 1993). For the equivalent form with denominator , set , , and equal to the real cube root of . The class number examples use th-degree algebraic integers for these constants (Borwein and Borwein 1993). Among this family's class number 1 imaginary quadratic fields, the fastest convergence per term corresponds to the discriminant , the largest in absolute value. The corresponding series was given by Chudnovsky and Chudnovsky (1988). The 163 appearing here is the same one appearing in the fact that (the Ramanujan constant) is very nearly an integer. Similarly, the factor comes from the j-function identity for . The series is given by

(Borwein and Borwein 1993; Beck and Trott; Bailey et al. 2007, p. 44). This series gives 14 decimal digits accurately per term. The same equation in another form was given by the Chudnovsky brothers (1988) and is used by the Wolfram Language to calculate (Vardi 1991; Wolfram Research),

where , , and .

The class number 2 example with discriminant is

where

(Borwein and Borwein 1993). This series adds about 25 decimal digits for each additional term. The fastest converging series for class number 3 corresponds to and gives 37-38 decimal digits per term. The class number 4 example computed by Borwein and Borwein (1993) corresponds to and is

where

This gives 50 decimal digits per term. Borwein and Borwein (1993) have developed a general algorithm for generating such series for arbitrary class number.

A complete listing of Ramanujan's series for found in his second and third notebooks is given by Berndt (1994, pp. 352-354),

These equations were first proved by Borwein and Borwein (1987a, pp. 177-187). Borwein and Borwein (1987b, 1988, 1993) proved other equations of this type, and Chudnovsky and Chudnovsky (1988) found similar equations for other transcendental constants (Bailey et al. 2007, pp. 44-45).

Some independent equations of this type are given by

for with nonalternating signs,

for with alternating signs,

for (Guillera 2002, 2003, 2006),

for . The last identity was discovered by B. Gourevitch and reported by Guillera (2003); Au (2025) proved it, and Guillera (2026) gave a simpler proof.

For , J. Cullen (pers. comm., Dec. 8, 2010) discovered the series

which was proved by Guillera (2026).

Bailey et al. (2007, pp. 45-48) found no additional identities for or 5 in their restricted search. More general series for have since been proved by Au (2025).

Bellard (1997b) found the exotic formula numerically using the PSLQ algorithm

where

Gasper quotes the result

where is a generalized hypergeometric function, and transforms it to

A fascinating result due to Gosper is given by

satisfies the inequality

D. Terr (pers. comm.) noted the curious identity

involving the first 9 digits of pi.