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...)