[size=150]There are many formulas of [img width=7,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline1.gif[/img] of many types. Among others, these include series, products, geometric constructions, limits, special values, and [url=https://mathworld.wolfram.com/PiIterations.html]pi iterations[/url].[img width=7,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline2.gif[/img] is intimately related to the properties of circles and spheres. For a circle of [url=https://mathworld.wolfram.com/Radius.html]radius[/url] [img width=6,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline3.gif[/img], the circumference and area are given by[br][br][table][tr][td][img width=10,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline4.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline5.gif[/img][/td][td][img width=26,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline6.gif[/img][/td][td][br][/td][/tr][tr][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline7.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline8.gif[/img][/td][td][img width=26,height=17]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline9.gif[/img][/td][td][br][/td][/tr][/table][br]Similarly, for a sphere of [url=https://mathworld.wolfram.com/Radius.html]radius[/url] [img width=6,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline10.gif[/img], the surface area and volume enclosed are[br][br][table][tr][td][img width=8,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline11.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline12.gif[/img][/td][td][img width=32,height=17]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline13.gif[/img][/td][td][br][/td][/tr][tr][td][img width=10,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline14.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline15.gif[/img][/td][td][img width=39,height=23]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline16.gif[/img][/td][td][br][/td][/tr][/table][br]An exact formula for [img width=7,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline17.gif[/img] in terms of the [url=https://mathworld.wolfram.com/InverseTangent.html]inverse tangents[/url] of [url=https://mathworld.wolfram.com/UnitFraction.html]unit fractions[/url] is [url=https://mathworld.wolfram.com/MachinsFormula.html]Machin's formula[/url][table][tr][td][img width=174,height=23]https://mathworld.wolfram.com/images/equations/PiFormulas/NumberedEquation1.gif[/img][/td][td][br][/td][/tr][/table][br]There are three other [url=https://mathworld.wolfram.com/Machin-LikeFormulas.html]Machin-like formulas[/url], as well as thousands of other similar formulas having terms.[br][br][img width=405,height=250]https://mathworld.wolfram.com/images/eps-gif/GregorySeries_1000.gif[/img][br][br][br]Gregory and Leibniz found[table][tr][td][img width=12,height=32]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline18.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline19.gif[/img][/td][td][img width=67,height=46]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline20.gif[/img][/td][td][br][/td][/tr][tr][td][img width=12,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline21.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline22.gif[/img][/td][td][img width=89,height=35]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline23.gif[/img][/td][td][br][/td][/tr][/table][br](Wells 1986, p. 50), which is known as the [url=https://mathworld.wolfram.com/GregorySeries.html]Gregory series[/url] and may be obtained by plugging [img width=31,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline24.gif[/img] into the [url=https://mathworld.wolfram.com/LeibnizSeries.html]Leibniz series[/url] for [img width=39,height=17]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline25.gif[/img]. The error after the [img width=7,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline26.gif[/img]th term of this series in the [url=https://mathworld.wolfram.com/GregorySeries.html]Gregory series[/url] is larger than [img width=37,height=17]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline27.gif[/img] so this sum converges so slowly that 300 terms are not sufficient to calculate [img width=7,height=15]https://mathworld.wolfram.com/images/equations/PiFormulas/Inline28.gif[/img] correctly to two decimal places! However, it can be transformed to[table][tr][td][br][img width=138,height=46]https://mathworld.wolfram.com/images/equations/PiFormulas/NumberedEquation2.gif[/img][/td][/tr][/table][br][br][img width=154,height=146]https://mathworld.wolfram.com/images/eps-gif/CirclePi_1000.gif[/img][br][br]The constant pi, denoted [img width=7,height=15]https://mathworld.wolfram.com/images/equations/Pi/Inline1.gif[/img], is a [url=https://mathworld.wolfram.com/RealNumber.html]real number[/url] defined as the ratio of a [url=https://mathworld.wolfram.com/Circle.html]circle[/url]'s [url=https://mathworld.wolfram.com/Circumference.html]circumference[/url] [img width=10,height=15]https://mathworld.wolfram.com/images/equations/Pi/Inline2.gif[/img] to its [url=https://mathworld.wolfram.com/Diameter.html]diameter[/url] [img width=41,height=15]https://mathworld.wolfram.com/images/equations/Pi/Inline3.gif[/img],[table][tr][td][img width=7,height=15]https://mathworld.wolfram.com/images/equations/Pi/Inline4.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/Pi/Inline5.gif[/img][/td][td][img width=15,height=35]https://mathworld.wolfram.com/images/equations/Pi/Inline6.gif[/img][/td][td][br][/td][/tr][tr][td][img width=12,height=15]https://mathworld.wolfram.com/images/equations/Pi/Inline7.gif[/img][/td][td][img width=9,height=15]https://mathworld.wolfram.com/images/equations/Pi/Inline8.gif[/img][/td][td][img width=21,height=35]https://mathworld.wolfram.com/images/equations/Pi/Inline9.gif[/img][/td][td][br][/td][/tr][/table][img width=7,height=15]https://mathworld.wolfram.com/images/equations/Pi/Inline10.gif[/img] has decimal expansion given by[table][tr][td][img width=325,height=15]https://mathworld.wolfram.com/images/equations/Pi/NumberedEquation1.gif[/img][br][br][/td][td][/td][/tr][/table][/size][table][tr][td][br][br][/td][/tr][/table]