# Approximations of π

An approximation of π is a numerical value, rational number, or series used in place of the exact ratio of a circle's circumference to its diameter, which is an irrational number and cannot be written out in full. The history of the constant is largely a history of the methods used to bound and compute it, from geometric polygon bounds in antiquity to series and algorithms that now produce hundreds of trillions of digits.<sup>[1](https://www.researchgate.net/publication/398124726_The_Infinite_Quest_A_Comprehensive_Survey_of_Approximations_to_p)</sup>

| Key fact | Detail |
|---|---|
| Archimedes' bounds | 223/71 < π < 22/7, proved around 250 BC with inscribed and circumscribed polygons of up to 96 sides<sup>[2](https://www.qatar.cmu.edu/wp-content/uploads/2025/09/FinalRoundQuestions2016.pdf)</sup> |
| Quality of 22/7 | Accurate to one part in 2484; the reported Babylonian approximation 25/8 (c. 2000 BC) is accurate to one part in 189<sup>[3](https://www.cs.miami.edu/home/burt/manuscripts/archimedes/archimedes_pi.html)</sup> |
| Name | π is sometimes called "Archimedes' constant" because his polygonal algorithm dominated for over 1,000 years<sup>[2](https://www.qatar.cmu.edu/wp-content/uploads/2025/09/FinalRoundQuestions2016.pdf)</sup> |
| First improvement | Ptolemy's sexagesimal value 3° 8′ 30″ was the first approximation more exact than Archimedes'<sup>[4](https://en.wikisource.org/wiki/%22Squaring_the_Circle%22%3A_a_History_of_the_Problem/Chapter_2)</sup> |
| Series era | Madhava of Sangamagrama (c. 1400) found a series for π, rediscovered by Leibniz in 1674; the Chudnovsky algorithm followed in 1988<sup>[1](https://www.researchgate.net/publication/398124726_The_Infinite_Quest_A_Comprehensive_Survey_of_Approximations_to_p)</sup> |
| Current record | 314 trillion digits, computed by StorageReview in a 110-day run on a single server, announced December 2025<sup>[5](https://www.tomshardware.com/pc-components/storage/pi-calculating-record-shattered-at-314-trillion-digits-with-a-four-month-run-on-a-single-server-storagereview-retakes-the-crown-thanks-to-storage-bandwidth)</sup> |

## Archimedes and the polygon method

The first recorded algorithm for rigorously calculating π was a geometrical approach using polygons, devised around 250 BC by Archimedes of Syracuse (287–212 BC). He drew a regular hexagon inside and outside a circle and successively doubled the number of sides until he reached a 96-sided regular polygon. From the perimeters of these polygons he derived the bounds 223/71 < π < 22/7 — in other words, 3.1408 < π < 3.1429.<sup>[2](https://www.qatar.cmu.edu/wp-content/uploads/2025/09/FinalRoundQuestions2016.pdf)</sup><sup> • </sup><sup>[6](https://uregina.ca/~kozdron/Research/UgradTalks/PiDay.pdf)</sup>

In modern notation, if \( P_{n} \) is the semi-perimeter of a regular n-gon circumscribed about a unit circle and \( p_{n} \) the semi-perimeter of the inscribed n-gon, then \( P_{n} = n/\cot(\pi/n) \) and \( p_{n} = n/\csc(\pi/n) \), and with increasing n the two bounds close in on the true value: \( p_{n} < \pi < P_{n} \).<sup>[3](https://www.cs.miami.edu/home/burt/manuscripts/archimedes/archimedes_pi.html)</sup> [Archimedes](https://www.edgepedia.org/archimedes) knew that π does not equal 22/7 and made no claim to have found its exact value; the upper bound is simply a very good rational approximation.<sup>[7](https://mathshistory.st-andrews.ac.uk/HistTopics/Pi_through_the_ages/)</sup>

The quality of 22/7 is unusual for so small a fraction. Archimedes' overestimate was used for hundreds of years afterwards by Greek mathematicians and by mathematicians of the Arab Empire.<sup>[8](https://bulldog2.redlands.edu/fac/beery/math115/m115_activ_est_pi.htm)</sup>

## Ancient and medieval refinements

A reported Babylonian approximation from around 2000 BC is π ≈ 25/8, good to one part in 189.<sup>[3](https://www.cs.miami.edu/home/burt/manuscripts/archimedes/archimedes_pi.html)</sup> Archimedes' result was the first in history correct to two places after the decimal point.<sup>[8](https://bulldog2.redlands.edu/fac/beery/math115/m115_activ_est_pi.htm)</sup>

Around 250 CE the Chinese mathematician [Liu Hui](https://www.edgepedia.org/liu-hui), probably unaware of Archimedes' work, used an inscribed polygon with 192 sides to obtain a slightly better approximation than Archimedes had made.<sup>[8](https://bulldog2.redlands.edu/fac/beery/math115/m115_activ_est_pi.htm)</sup> The best approximation of π preserved from Greco-Roman antiquity appears in Ptolemy in the early 100s CE, expressed in sexagesimal measure as 3° 8′ 30″, the first value more exact than Archimedes'. The same value in the form 3927/1250 was given by Bhāskara, born 1114 in his work *The crowning of the system*; he described it as exact, in contrast with the inexact 22/7.<sup>[4](https://en.wikisource.org/wiki/%22Squaring_the_Circle%22%3A_a_History_of_the_Problem/Chapter_2)</sup>

## Series and arctangent formulas

A turning point came with infinite series. A series for π was discovered by [Madhava of Sangamagrama](https://www.edgepedia.org/madhava-of-sangamagrama) around 1400 and rediscovered by Leibniz in 1674.<sup>[1](https://www.researchgate.net/publication/398124726_The_Infinite_Quest_A_Comprehensive_Survey_of_Approximations_to_p)</sup>

Modern high-precision computation rests on hypergeometric series. Many of the fastest known algorithms for computing π involve generalized hypergeometric series such as the [Ramanujan–Sato series](https://www.edgepedia.org/ramanujan-sato-series), and Ramanujan's own formula for 1/π converges rapidly enough to underpin this work.<sup>[9](https://arxiv.org/abs/2108.00844)</sup><sup> • </sup><sup>[10](https://planetmath.org/RamanujansFormulaForPi)</sup> The Chudnovsky algorithm, introduced in 1988, is one such series-based method used in record computations.<sup>[1](https://www.researchgate.net/publication/398124726_The_Infinite_Quest_A_Comprehensive_Survey_of_Approximations_to_p)</sup>

## Digit records in the computing era

Calculating digits of π began as an informal pursuit and became a way to benchmark the floating-point performance of CPUs; as the calculations grew, RAM, I/O architecture, and storage systems became decisive.<sup>[5](https://www.tomshardware.com/pc-components/storage/pi-calculating-record-shattered-at-314-trillion-digits-with-a-four-month-run-on-a-single-server-storagereview-retakes-the-crown-thanks-to-storage-bandwidth)</sup> Google Cloud computed 100 trillion digits in 2022.<sup>[11](https://www.techradar.com/pro/anyone-for-a-slice-of-record-pi-new-landmark-sees-314-trillion-digits-calculated-as-news-site-trounces-google-cloud-for-now)</sup> In early 2024, StorageReview reached the 105 trillionth digit and later improved this to 202 trillion.<sup>[12](https://www.guinnessworldrecords.com/news/commercial/2026/6/pi-high-computing-achievement-technology-companies-calculate-pi-to-340-trillionth-digit)</sup> A 300-trillion-digit record by Linus Media Group (Canada) and KIOXIA (USA) was accomplished on 2 April of the following year.<sup>[12](https://www.guinnessworldrecords.com/news/commercial/2026/6/pi-high-computing-achievement-technology-companies-calculate-pi-to-340-trillionth-digit)</sup>

In December 2025 StorageReview set a new record of 314 trillion digits on a single on-premises server, a Dell PowerEdge R7725 with two AMD Epyc 192-core chips (384 cores total) and 1.5 TB of DDR5 memory. The run lasted 110 days, shorter than the roughly 225 days of the previous large-scale record, and used the y-cruncher application with memory and storage tuning matched to its behavior.<sup>[5](https://www.tomshardware.com/pc-components/storage/pi-calculating-record-shattered-at-314-trillion-digits-with-a-four-month-run-on-a-single-server-storagereview-retakes-the-crown-thanks-to-storage-bandwidth)</sup><sup> • </sup><sup>[11](https://www.techradar.com/pro/anyone-for-a-slice-of-record-pi-new-landmark-sees-314-trillion-digits-calculated-as-news-site-trounces-google-cloud-for-now)</sup>

## Why approximations still matter

For practical work, simple rationals remain useful. The value 22/7 retains interest because of its simplicity and can be used in practice when the precision sought is not too high.<sup>[13](https://accromath.uqam.ca/2012/07/regard-archimedien-sur-le-cercle-la-quete-du-fameux-227/)</sup> At the other extreme, π enters essentially every formula of spacecraft motion, materials science, and propulsion, and anything round or cyclical, such as radio waves, involves it.<sup>[14](https://apnews.com/article/549286e6ea0a093cbc75f3b17fdc150f)</sup> The digit records serve mainly as computation benchmarks rather than as sources of digits anyone needs; π is an exact number by definition, and approximation enters only when it is expressed in digital form.<sup>[15](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nbsspecialpublication300v1.pdf)</sup>

## References

1. [The Infinite Quest: A Comprehensive Survey of Approximations to π](https://www.researchgate.net/publication/398124726_The_Infinite_Quest_A_Comprehensive_Survey_of_Approximations_to_p)
2. [Approximating π, Concepts of Mathematics, Carnegie Mellon University Qatar](https://www.qatar.cmu.edu/wp-content/uploads/2025/09/FinalRoundQuestions2016.pdf)
3. [Archimedes and Pi, University of Miami](https://www.cs.miami.edu/home/burt/manuscripts/archimedes/archimedes_pi.html)
4. ["Squaring the Circle": a History of the Problem, Chapter 2, Wikisource](https://en.wikisource.org/wiki/%22Squaring_the_Circle%22%3A_a_History_of_the_Problem/Chapter_2)
5. [Pi calculation world record shattered at 314 trillion digits, Tom's Hardware](https://www.tomshardware.com/pc-components/storage/pi-calculating-record-shattered-at-314-trillion-digits-with-a-four-month-run-on-a-single-server-storagereview-retakes-the-crown-thanks-to-storage-bandwidth)
6. [Pi Day talk, University of Regina](https://uregina.ca/~kozdron/Research/UgradTalks/PiDay.pdf)
7. [A history of Pi, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Pi_through_the_ages/)
8. [Estimating π activity, University of Redlands](https://bulldog2.redlands.edu/fac/beery/math115/m115_activ_est_pi.htm)
9. [Asymptotic expansions for the truncation error in Ramanujan-type series, arXiv](https://arxiv.org/abs/2108.00844)
10. [Ramanujan's formula for pi, PlanetMath](https://planetmath.org/RamanujansFormulaForPi)
11. [New landmark sees 314 trillion digits calculated, TechRadar](https://www.techradar.com/pro/anyone-for-a-slice-of-record-pi-new-landmark-sees-314-trillion-digits-calculated-as-news-site-trounces-google-cloud-for-now)
12. [Pi-high computing achievement, Guinness World Records](https://www.guinnessworldrecords.com/news/commercial/2026/6/pi-high-computing-achievement-technology-companies-calculate-pi-to-340-trillionth-digit)
13. [Regard archimédien sur le cercle: la quête du fameux 22/7, Accromath](https://accromath.uqam.ca/2012/07/regard-archimedien-sur-le-cercle-la-quete-du-fameux-227/)
14. [From rockets to cancer research, here's how pi is embedded in our lives, AP News](https://apnews.com/article/549286e6ea0a093cbc75f3b17fdc150f)
15. [Precision measurement and calibration, NBS Special Publication 300](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nbsspecialpublication300v1.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Real and complex number constructions › Irrational numbers*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license