NASA uses a value of π = 3.141592653589793 with 15 decimal digits for interplanetary navigation.

Calculating the circumference of the circle with radius = 32 billion km, slightly larger than the distance of the Voyager 1 spacecraft, and π with 15 digits, gives an error of just 1.5 cm.

Calculating the circumference of the Universe with radius = 46 billion light years, with an accuracy equal to the diameter of a hydrogen atom, requires 37 decimal digits.
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https://www.jpl.nasa.gov/edu/news/how-many-decimals-of-pi-do-we-really-need/
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This remarkable equation for π by Indian mathematician Srinivasa Ramanujan, developed around 1910-1914, is an infinite series with some curious numbers and an interesting property. The first k terms of the sum give π with ~8*k decimal digits.

E.g., the 1st term gives π = 3.141592 73001..., which is accurate to 6 decimal places.

Using 2 terms, we get π = 3.141592 653589793 87808..., which is accurate to 15 decimal places.

https://www.piday.org/million/
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Over 75 years after Ramanujan developed the equation for π, the Chudnovsky brothers published this formula, which produces about 14 digits of π per term, more than 2x that of Ramanujan's formula.

There's some sorcery in these numbers.
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https://en.wikipedia.org/wiki/Chudnovsky_brothers
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@AkaSci The seventh equation was found in 1995.

https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula

BTW it occurs twice in your list. If it is in time-of-discovery order, the last one is the proper one.

BTW2 the first equation (Euler?) is just pretty. You cannot compute with it...