Intuitive

It's been bothering me since childhood, this perimeter
of the circle business, equal to 2*pi*r. In point of fact, for
the circle to measure 3... something, the radius of the circle
has to be 1/2!! This is the set up that initiates Viète's pi formula.

This simplifies to a frrst term of (1/2)^.5

For the second term - and this is the work of the formula - Viète gives

a reading of the perimeter of an octagon. 

This is the geometric intuition to the formula. 

                                            *     *     *

Asked for 50 loops, 30 digits:

Asked for 50 loops, 34 digits:

*     *     *