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Pi Approximations

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Pi approximations are expressions whose values are close to pi. Throughout this entry, accuracy to d decimal digits means that the absolute error is less than 10^(-d). The simplest pi approximations are the convergents of the pi continued fractions. The first few are given by 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.

Two approximations follow from the near-identity function 3sinx/(2+cosx) evaluated at x=pi/4 and pi/8, giving

pi approx (12)/7(2sqrt(2)-1)
(1)
approx (24sqrt(2-sqrt(2)))/(4+sqrt(2+sqrt(2))),
(2)

which are good to 2 and 3 decimal digits, respectively. Kochanski's approximation is the root of

9x^4-240x^2+1492=0
(3)

given by

pi approx sqrt((40)/3-sqrt(12)) approx 3.141533,
(4)

which is good to 4 decimal digits.

Other approximations include

pi approx 2sqrt(6)+3sqrt(2)-6
(5)
approx sqrt(231-sqrt(3))-12
(6)

(P. B. S. Bueser, pers. comm., Sep. 25, 2026), which are good to 4 and 5 decimal digits, respectively.

A simple power approximation is

pi approx (7^7)/(8^6)=3.1415672302...,
(7)

which is good to 4 decimal digits.

E. Weisstein (using Wolfram|Alpha, Oct. 29, 2026) obtained the approximations

pi approx (-143e^2+133e+8)/(5(2e^2-23e+4))
(8)
approx [3/2(5e+5+ln8)]^(1/3)
(9)
approx (133-37sqrt(5))/(16),
(10)

which are good to 14, 7, and 6 decimal digits, respectively. The same search gave the root x=3.14159265382964... of

37x^3-114x^2-36x+91=0,
(11)

which approximates pi to 9 decimal digits.

An approximation involving Apéry's constant zeta(3) is

pi approx (ln(2941/25))/(sqrt(68)ln[zeta(3)]),
(12)

which is good to 5 decimal digits (V. Muleva, pers. comm., Sep. 17, 2022).

Another curious fact is the almost integer

e^pi-pi=19.999099979...,
(13)

which can also be written as

(pi+20)^i=-0.9999999992-0.0000388927i approx -1
(14)
cos(ln(pi+20)) approx -0.9999999992.
(15)

Here, e^pi is Gelfond's constant. Applying cosine a few more times gives

cos(picos(picos(ln(pi+20)))) approx -1+3.9321609261×10^(-35).
(16)

Another approximation involving e is given by

pi approx sqrt(4e-1),
(17)

which is good to 3 decimal digits (A. Povolotsky, pers. comm., Mar. 6, 2008).

An apparently interesting near-identity is given by

sin(1/(555555) degrees) approx pi×10^(-8),
(18)

which becomes less surprising when it is noted that 555555 is a repdigit, so the above is just a special case of the near-identity

sin((pi/(180))/(d(10^n-1)/9)) approx sin((pi/(180))/(d(10^n)/9))
(19)
approx pi/(2d)10^(-(n+1))
(20)

with d=5 and n=6.

An approximation involving the golden ratio phi arises from the volume of the canonical tetragonal antiwedge with unit midradius, namely

pi approx 8/3phi(2-sqrt(phi))
(21)
=8/3(1+sqrt(5)-sqrt(2+sqrt(5)))
(22)
=3.14105...
(23)

(cf. Pegg 2018), which is good to 3 decimal digits. Another approximation involving phi is

pi approx 6/5phi^2
(24)
=6/5((sqrt(5)+1)/2)^2
(25)
=3/5(3+sqrt(5))
(26)
=3.14164...,
(27)

which is good to 4 decimal digits. A similar approximation due to S. Mircea-Mugurel (pers. comm., Oct. 30, 2002) is given by

pi approx 4phi^(-1/2)=3.1446...,
(28)

which however is only good to two decimal places. Another approximation involving the golden ratio phi is given by

pi approx ((802phi-801)/(602phi-601))^4,
(29)

which is good to 7 decimal digits (K. Rashid, pers. comm.).

Some approximations due to Ramanujan include

pi approx (19sqrt(7))/(16)
(30)
approx 7/3(1+1/5sqrt(3))
(31)
approx 9/5+sqrt(9/5)
(32)
approx ((2143)/(22))^(1/4)=(9^2+(19^2)/(22))^(1/4)=(102-(2222)/(22^2))^(1/4)=(97+1/2-1/(11))^(1/4)=(97+9/(22))^(1/4)
(33)
approx (63)/(25)((17+15sqrt(5))/(7+15sqrt(5)))
(34)
approx (355)/(113)(1-(0.0003)/(3533))
(35)
approx (12)/(sqrt(130))ln[((3+sqrt(13))(sqrt(8)+sqrt(10)))/2]
(36)
approx (24)/(sqrt(142))ln[(sqrt(10+11sqrt(2))+sqrt(10+7sqrt(2)))/2]
(37)
approx (12)/(sqrt(190))ln[(3+sqrt(10))(sqrt(8)+sqrt(10))]
(38)
approx (12)/(sqrt(310))ln[1/4(3+sqrt(5))(2+sqrt(2))(5+2sqrt(10)+sqrt(61+20sqrt(10)))]
(39)
approx 4/(sqrt(522))ln[((5+sqrt(29))/(sqrt(2)))^3(5sqrt(29)+11sqrt(6))(sqrt((9+3sqrt(6))/4)+sqrt((5+3sqrt(6))/4))^6],
(40)

whose absolute errors are less than 10^(-d) for d=3, 4, 4, 8, 9, 14, 15, 15, 18, 23, 31, respectively (Ramanujan 1913-1914; Hardy 1952, p. 70; Wells 1986, p. 54; Berndt 1994, pp. 48-49 and 88-89). Equation (33) and the similar

pi approx (66sqrt(2))/(33sqrt(29)-148)
(41)

are also given by Borwein and Bailey (2003, p. 135). Ramanujan also gave

pi approx (99^2)/(2206sqrt(2))
(42)

(Wells 1986, p. 54).

Approximations to pi can be formed as ratios of two 10-digit pandigital numbers, each using every digit from 0 through 9 exactly once (A. Povolotsky, pers. comm., Aug. 29, 2022). The closest such approximation is

pi approx (8405139762)/(2675439081)=3.141592653591...
(43)

which approximates pi to 11 decimal digits (E. Weisstein, Sep. 7, 2022). S. Irvine (pers. comm.) noted that (33), giving an approximation to pi good to 8 decimal digits, can be written in a pandigital form as

pi approx 0+sqrt(sqrt(3^4+(19^2)/(78-56)))
(44)
=(9^2+(19^2)/(22))^(1/4)
(45)
=((2143)/(22))^(1/4)
(46)

(S. Plouffe, pers. comm.; cf. Wells 1986, p. 54). E. Pegg (pers. comm.) found the pandigital approximation

0+3+(1-(9-8^(-5))^(-6))/(7+2^(-4))=(233546921420225777694970883318153571)/(74340293968115785654927455866388593)
(47)

which approximates pi to 9 decimal digits. Another pandigital formula is given by

pi approx 3+4/(28)-1/(790+5/6)=3.14159265392...
(48)

(B. Astle, pers. comm., Jan. 9, 2004), which approximates pi to 9 decimal digits. Surpassing both of these is the pandigital approximation

2^(5^(.4))-.6-((.3^9)/7)^(.8^(.1)).
(49)

which is accurate to 12 decimal digits (B. Ziv, pers. comm., Jul. 7, 2004). A further pandigital approximation is given by

(ln{[2×5!+(8-1)!]^(sqrt(9))+4!+(3!)!})/(sqrt(67)),
(50)

which is good to 17 decimal digits (G. W. Barbosa, pers. comm.).

M. Schneider (pers. comm., May 6, 2008) found the approximation

pi approx sqrt(7+sqrt(6+sqrt(5))),
(51)

which is good to 4 decimal digits. P. Lindborg (pers. comm.) noted that the convergent 104348/33215 can be written in the slightly curious form

(314+142)/(2·3·5·7)(1373)/(13·73),
(52)

which is good to 9 decimal digits.

Other approximations due to E. Pegg include

pi approx 4-((105)/(166))^(1/3),
(53)

which is good to 6 decimal digits (pers. comm., Mar. 2, 2002) and

pi approx (22)/(17)+(37)/(47)+(88)/(83),
(54)

which is good to 9 decimal digits (pers. comm., Dec. 30, 2002).

A simple approximation involving the cube root is

pi approx 31^(1/3),
(55)

which is good to 3 decimal digits (M. Joseph, pers. comm., May 3, 2006).

A difference of two cube roots gives

pi approx 318^(1/3)-50^(1/3),
(56)

which is good to 7 decimal digits (F. Franco, pers. comm., May 14, 2016).

A more exotic approximation is given by

pi approx (ln6)^((ln5)^((ln4)^((ln3)^(ln2)))),
(57)

which is good to 4 decimal digits (M. Joseph, pers. comm., May 3, 2006).

Castellanos (1988ab) gives a slew of curious formulas:

pi approx (2e^3+e^8)^(1/7)
(58)
approx ((553)/(311+1))^2
(59)
approx (3/(14))^4((193)/5)^2
(60)
approx ((296)/(167))^2
(61)
approx ((66^3+86^2)/(55^3))^2
(62)
approx 1.09999901·1.19999911·1.39999931·1.69999961
(63)
approx (47^3+20^3)/(30^3)-1
(64)
approx 2+sqrt(1+((413)/(750))^2)
(65)
approx ((77729)/(254))^(1/5)
(66)
approx (31+(62^2+14)/(28^4))^(1/3)
(67)
approx (1700^3+82^3-10^3-9^3-6^3-3^3)/(69^5)
(68)
approx (95+(93^4+34^4+17^4+88)/(75^4))^(1/4)
(69)
approx (100-(2125^3+214^3+30^3+37^2)/(82^5))^(1/4),
(70)

which are accurate to 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, and 13 decimal digits, respectively. An extremely accurate approximation due to Shanks (1982) is

pi approx 6/(sqrt(3502))ln(2u)+7.37×10^(-82),
(71)

where u is the product of four simple quartic units.

David W. Hoffman (pers. comm.) gave

pi approx ((10^(100))/(11222.11122))^(1/193),
(72)

where the numerator is one googol. This approximation is good to 10 decimal digits. The approximations

pi approx e^(e^(e^(-2)))
(73)
approx 2+e^(e^(-2))
(74)

give 3 and 2 decimal digits, respectively (G. von Hippel, pers. comm.).

A sequence of approximations due to Plouffe and Borwein and Bailey (2003, pp. 115 and 134-135) includes

pi approx 43^(7/23)
(75)
approx (ln2198)/(sqrt(6))
(76)
approx ((13)/4)^(1181/1216)
(77)
approx (689)/(396ln((689)/(396)))
(78)
approx ln5280sqrt(9/(67))
(79)
approx ((63023)/(30510))^(1/3)+1/4+1/2(sqrt(5)+1)
(80)
approx (48)/(23)ln((60318)/(13387))
(81)
approx (228+(16)/(1329))^(1/41)+2
(82)
approx (125)/(123)ln((28102)/(1277))
(83)
approx 3/(sqrt(163))ln(640320)
(84)
approx ((276694819753963)/(226588))^(1/158)+2
(85)
approx (ln(640320^3+744))/(sqrt(163)),
(86)

which are accurate to 4, 5, 7, 7, 9, 10, 11, 11, 11, 15, 23, and 30 decimal digits, respectively.

The last expression follows from the series expansion of the j-function. Carrying this one step further gives

-e^(pisqrt(163))+744-196884e^(-pisqrt(163))+...=-640320^3
(87)
e^(pisqrt(163))(1+196884e^(-2pisqrt(163))) approx 640320^3+744
(88)
e^(2pisqrt(163))(1+196884e^(-2pisqrt(163)))^2 approx (640320^3+744)^2
(89)
e^(2pisqrt(163))+2·196884 approx (640320^3+744)^2
(90)

giving

pi approx (ln[(640320^3+744)^2-2·196884])/(2sqrt(163)),
(91)

which is good to 46 decimal digits (Warda, pers. comm., Nov. 15, 2004).

PiApproximationsSqrt

Interestingly, ln(nint(exp(pisqrt(163n))))/sqrt(163n) gives successively good approximations to pi for larger and larger n (Warda, pers. comm., Nov. 22, 2004). In particular, the number of correct decimal digits for n=1, 2, ... are given by 30, 28, 31, 46, 40, 44, 48, 51, 61, 57, 59, 62, 65, ... (OEIS A100935).

An approximation due to Stoschek using powers of two and the special number 163 (the largest Heegner number) is given by

pi approx (2^9)/(163)=(512)/(163) approx 3.1411043,
(92)

which is good to 3 decimal digits. A fraction with small numerator and denominator which gives a close approximation to pi is

(311)/(99)=3.14141414....
(93)

Some approximations involving the ninth roots of rational numbers include

pi approx ((4297607660)/(144171))^(1/9)
(94)
approx ((34041350274878)/(1141978491))^(1/9),
(95)

which are good to 12 and 15 decimal digits, respectively (P. Galliani, pers. comm.).

de Jerphanion (pers. comm.) found

pi approx ln(23+1/(22)+2/(21))=ln(23+1/6-2/(77))=ln((10691)/(462)),
(96)

which is good to 9 decimal digits, and J. Iuliano found

pi approx ((19)/(60)+1/(sqrt(3·123449)))^(-1),
(97)

which is good to 11 decimal digits.

Definite integrals giving approximations to pi were considered by Backhouse (1995) and Lucas (2005).

F. Voormanns (pers. comm., Dec. 12, 2003) found the curious astronomical approximation

pi approx 3+(13Y-42)/(91Y),
(98)

where Y is the numerical length of the year in days and a week is seven days. This is accurate to 9 decimal digits for Y=365, or 6 decimal digits for the average Gregorian year (Y=365.2425) or tropical year (Y=365.242190).

Rivera gives other approximation formulas.

See also

Almost Integer, Pi, Pi Continued Fraction, Pi Digits, Pi Formulas

Portions of this entry contributed by Percival Byron S. Bueser

Explore with Wolfram|Alpha

References

Backhouse, N. "Note 79.36. Pancake Functions and Approximations to pi." Math. Gaz. 79, 371-374, 1995.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988a.Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148-163, 1988b.Contest Center. "Pi Competition." https://www.contestcen.com/pi.htm.Friedman, E. "Problem of the Month (August 2004)." https://erich-friedman.github.io/mathmagic/0804.html.Hardy, G. H. A Course of Pure Mathematics, 10th ed. Cambridge, England: Cambridge University Press, 1952.Lucas, S. K. "Integral Proofs that 355/113>pi." Gaz. Austral. Math. Soc. 32, 263-266, 2005.Pegg, E. Jr. "For Pi Day: Volume = 3.141--The Canonical Tetragonal Antiwedge Hexahedron." Mar. 14, 2018. https://community.wolfram.com/t/12115.Plouffe, S. "A Few Approximations of Pi." https://web.archive.org/web/20030228230356/http://pi.lacim.uqam.ca/eng/approximations_en.html.Ramanujan, S. "Modular Equations and Approximations to pi." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.Rivera, C. "Problems & Puzzles: Puzzle 050-The Best Approximation to Pi with Primes." https://www.primepuzzles.net/puzzles/puzz_050.htm.Shanks, D. "Dihedral Quartic Approximations and Series for pi." J. Number. Th. 14, 397-423, 1982.Sloane, N. J. A. Sequences A002485/M3097, A002486/M4456, A100935, and A114526 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.

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Pi Approximations

Cite this as:

Weisstein, Eric W., with contributions by Percival Byron S. Bueser. "Pi Approximations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PiApproximations.html

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