List of formulae involving π
Updated: 5/24/2026, 6:54:05 PM Wikipedia source
The following is a list of significant formulae involving the mathematical constant π. Many of these formulae can be found in the article Pi, or the article Approximations of π.
Tables
· Formulae yielding <i>π</i> › Other infinite series
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
6
n
+
1
)
(
π
=
32
Z
{\displaystyle \pi ={\frac {32}{Z}}}
π
=
32
Z
{\displaystyle \pi ={\frac {32}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
32
Z
{\displaystyle \pi ={\frac {32}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
5
−
π
=
27
4
Z
{\displaystyle \pi ={\frac {27}{4Z}}}
π
=
27
4
Z
{\displaystyle \pi ={\frac {27}{4Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
27
4
Z
{\displaystyle \pi ={\frac {27}{4Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
2
27
)
n
π
=
15
3
2
Z
{\displaystyle \pi ={\frac {15{\sqrt {3}}}{2Z}}}
π
=
15
3
2
Z
{\displaystyle \pi ={\frac {15{\sqrt {3}}}{2Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
15
3
2
Z
{\displaystyle \pi ={\frac {15{\sqrt {3}}}{2Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
4
125
)
n
π
=
85
85
18
3
Z
{\displayst
π
=
85
85
18
3
Z
{\displayst
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
85
85
18
3
Z
{\displayst
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
4
85
)
n
π
=
5
5
2
3
Z
{\displaystyle
π
=
5
5
2
3
Z
{\displaystyle
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
5
5
2
3
Z
{\displaystyle
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
4
125
)
n
π
=
2
3
Z
{\displaystyle \pi ={\frac {2{\sqrt {3}}}{Z}}}
π
=
2
3
Z
{\displaystyle \pi ={\frac {2{\sqrt {3}}}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
2
3
Z
{\displaystyle \pi ={\frac {2{\sqrt {3}}}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
8
n
+
1
)
(
π
=
3
9
Z
{\displaystyle \pi ={\frac {\sqrt {3}}{9Z}}}
π
=
3
9
Z
{\displaystyle \pi ={\frac {\sqrt {3}}{9Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
3
9
Z
{\displaystyle \pi ={\frac {\sqrt {3}}{9Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
40
n
+
3
)
(
π
=
2
11
11
Z
{\displaystyle \pi ={\frac {2{\sqrt {11}}}{11Z}}}
π
=
2
11
11
Z
{\displaystyle \pi ={\frac {2{\sqrt {11}}}{11Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
2
11
11
Z
{\displaystyle \pi ={\frac {2{\sqrt {11}}}{11Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
280
n
+
19
)
(
π
=
2
4
Z
{\displaystyle \pi ={\frac {\sqrt {2}}{4Z}}}
π
=
2
4
Z
{\displaystyle \pi ={\frac {\sqrt {2}}{4Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
2
4
Z
{\displaystyle \pi ={\frac {\sqrt {2}}{4Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
10
n
+
1
)
(
π
=
4
5
5
Z
{\displaystyle \pi ={\frac {4{\sqrt {5}}}{5Z}}}
π
=
4
5
5
Z
{\displaystyle \pi ={\frac {4{\sqrt {5}}}{5Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
5
5
Z
{\displaystyle \pi ={\frac {4{\sqrt {5}}}{5Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
644
n
+
41
)
(
π
=
4
3
3
Z
{\displaystyle \pi ={\frac {4{\sqrt {3}}}{3Z}}}
π
=
4
3
3
Z
{\displaystyle \pi ={\frac {4{\sqrt {3}}}{3Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
3
3
Z
{\displaystyle \pi ={\frac {4{\sqrt {3}}}{3Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
π
=
72
Z
{\displaystyle \pi ={\frac {72}{Z}}}
π
=
72
Z
{\displaystyle \pi ={\frac {72}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
72
Z
{\displaystyle \pi ={\frac {72}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
π
=
3528
Z
{\displaystyle \pi ={\frac {3528}{Z}}}
π
=
3528
Z
{\displaystyle \pi ={\frac {3528}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
3528
Z
{\displaystyle \pi ={\frac {3528}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
| π = 1 {Z}}} | Z = ∑ n = 0 ∞ ( ( 2 n ) ! ) 3 |
| π = 4 {Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( |
| π = 4 {Z}}} | Z = ∑ n = 0 ∞ ( 6 n + 1 ) ( |
| π = 32 {Z}}} | Z = ∑ n = 0 ∞ ( 5 − |
| π = 27 4 {4Z}}} | Z = ∑ n = 0 ∞ ( 2 27 ) n |
| π = 15 3 2 }}{2Z}}} | Z = ∑ n = 0 ∞ ( 4 125 ) n |
| π = 85 85 18 3 Z {\displayst | Z = ∑ n = 0 ∞ ( 4 85 ) n |
| π = 5 5 2 3 Z {\displaystyle | Z = ∑ n = 0 ∞ ( 4 125 ) n |
| π = 2 3 }}{Z}}} | Z = ∑ n = 0 ∞ ( 8 n + 1 ) ( |
| π = 3 9 }{9Z}}} | Z = ∑ n = 0 ∞ ( 40 n + 3 ) ( |
| π = 2 11 11 }}{11Z}}} | Z = ∑ n = 0 ∞ ( 280 n + 19 ) ( |
| π = 2 4 }{4Z}}} | Z = ∑ n = 0 ∞ ( 10 n + 1 ) ( |
| π = 4 5 5 }}{5Z}}} | Z = ∑ n = 0 ∞ ( 644 n + 41 ) ( |
| π = 4 3 3 }}{3Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( |
| π = 4 {Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( |
| π = 72 {Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( |
| π = 3528 {Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( |
References
- The relation μ 0 = 4 π ⋅ 10 − 7 N / A 2 {\displaystyle \m
- (integral form of arctan over its entire domain, giving the period of tan)
- The coefficients can be obtained by reversing the Puiseux series of z ↦ z ∑ n = 0
- The n {\displaystyle n} th root with the smallest positive principal argument is chosen.
- When n ∈ Q + {\displaystyle n\in \mathbb {Q} ^{+}} , this gives algebraic approximations t
- When n ∈ Q + {\displaystyle {\sqrt {n}}\in \mathbb {Q} ^{+}} , this gives algebrai
- Regular and Chaotic Dynamicshttps://www.maths.tcd.ie/~lebed/Galperin.%20Playing%20pool%20with%20pi.pdf
- Real and Complex Analysis
- "A000796 – OEIS"https://oeis.org/A000796
- NIST Handbook of Mathematical Functionshttp://dlmf.nist.gov/19.8.i
- π Unleashed
- Numbers, constants and computationhttp://numbers.computation.free.fr/Constants/Algorithms/nthdecimaldigit.pdf
- "Weisstein, Eric W. "Pi Formulas", MathWorld"http://mathworld.wolfram.com/PiFormulas.html
- Algebra, an Elementary Text-book: Part II
- The Number Pi
- Ramanujan's Theta Functions
- Introductio in analysin infinitorum
- Carl B. Boyer, A History of Mathematics, Chapter 21., pp. 488–489
- Introductio in analysin infinitorum
- "Summing inverse squares by euclidean geometry"http://www.math.chalmers.se/~wastlund/Cosmic.pdf