Circle:
π = C / 2 R
π = A / R²
Sphere:
π = S / 4 R²
π = 3 V / 4 R³
Sphere in n dimensions:
π = (S ∙ Γ(1/2 + n/2) / Rⁿ) ^ (2 /(1 + n))
π = (V ∙ Γ(1 + n/2) / Rⁿ) ^ (2 / n)
Infinite series:
π = 4 ∙ (1/1 − 1/3 + 1/5 − 1/7 + 1/9 − ...)
π² = 8 ∙ (1/1² + 1/3² + 1/5² + 1/7² + 1/9² + ...)
π³ = 32 ∙ (1/1³ − 1/3³ + 1/5³ − 1/7³ + 1/9³ − ...)
etc.
Continued fraction:
π = 4 / (1 + 1² /(3 + 2² /(5 + 3² /(7 + 4² /(9 + ... )))))
π = 4 / (1 + 1² /(2 + 3² /(2 + 5² /(2 + 7² /(2 + ... )))))
etc.
Wallis’s product:
π = 2 ∙ (2/1 ∙ 2/3) ∙ (4/3 ∙ 4/5) ∙ (6/5 ∙ 6/7) ∙ (8/7 ∙ 8/9) ∙ ...
Viète’s formula:
π = 2 ∙ 2/sqrt(2) ∙ 2/sqrt(2 + sqrt(2)) ∙ 2/sqrt(2 + sqrt(2 + sqrt(2))) ∙ ...
Euler’s series based on the primes:
π = 1 + 1/2 + 1/3 + 1/4 −1/5 + 1/6 + 1/7 + 1/8 + 1/9 −1/10 + 1/11 + 1/12 −1/13 + ...
where, if the denominator is a prime of the form 4n − 1, the sign is positive; if the denominator is 2 or a prime of the form 4n + 1, the sign is negative; for composite numbers, the sign is equal the product of the signs of its factors.
π = 4 ∙ 3/4 ∙ 5/4 ∙ 7/8 ∙ 11/12 ∙ 13/12 ∙ 17/16 ∙ 19/20 ∙ 23/24 ∙ 29/28 ∙ 31/32 ∙ ...
where the numerators are the odd primes and each denominator is the multiple of four nearest to the numerator.
Riemann zeta function:
π²/6 = ζ(2) = 1/1² + 1/2² + 1/3² + 1/4² + ...
π⁴/90 = ζ(4) = 1/1⁴ + 1/2⁴ + 1/3⁴ + 1/4⁴ + ...
etc.
(By the way, 1 + 2 + 3 + 4 + 5 + ... = −1/12.)
Area under a bell curve:
√π = Γ(½) = ∫ exp(−x²) dx
Stirling’s appoximation:
√π = n! exp(n) / sqrt(2n) nⁿ , n → ∞
Simple pendulums:
π ≈ T sqrt(g / L) / 2
at small amplitudes of θ₀
π = T M(cos(θ₀ /2)) sqrt(g / L) / 2
where M(x) is the arithmetic-geometric mean of 1 and x.
Coulomb’s law:
π = q₁ q₂ / 4 F ε₀ r²
Heisenberg’s uncertainty principle:
π ≥ h / (4 Δx Δp)
Einstein’s field equation of general relativity:
π = bla bla bla + bla bla bla bla − bla bla bla
Many, many more:
http://en.wikipedia.org/wiki/List_of_formulae_involving_π
http://mathworld.wolfram.com/PiFormulas.html
http://numbers.computation.free.fr/Constants/Pi/piSeries.html
But best of all...
Euler’s identity:
π = log(−1) / sqrt(−1)
There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “Surely the population has nothing to do with the circumference of the circle.” Naturally, we are inclined to smile about the simplicity of the classmate’s approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense.
—E. P. Wigner
