The Real Circular Constant
The ratio of the circumference of a circle to the perimeter of the square containing it
is a constant k (kappa) the same for all circles.
The ratio of the area of a circle to that of the square containing it is also a constant,
the same for all circles; in fact the same constant k.
To fifty places of decimals this circular constant is k =
0.78539 81633 97448 30961 56608 45819 87572 10492 92349 84377 (64...)
To my way of thinking k seems more physically meaningful than the usual
p = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 (58...)
= 4k.
In fact p (pi) is the ratio of the area of a circle to that of a square on a radius, or it is the ratio of the circumference to a diameter, neither of which are comparisons of like properties of circle and square.
Leibniz found the neat but slowly converging series:
k = 1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + ...
The above 50 places for k have been calculated from the above 50 places for p which were found on this site which gives it to 1000 places, which apparently Prof. A. C. Aitken could recite from memory: pi to 1000 places
A convincing case can also be made for the replacement of p by t
= 2p, as in The Tau Manifesto by Michael Hartl.
t = 6.28318 53071 79586 47692 52867 66559 00576 83943 38798 75021 (16...)