The answer to your first question is definitely yes. For the Earth, it comes out that the circumference is about an inch smaller than pi*diameter.
The answer to your second question is no. There's no need to make "pi" itself variable. It's well-defined as the ratio of the circumference to the diameter on a flat part of space. It just isn't equal to that ratio on other parts of space. Pi stays the same, but the circumference/diameter ratio depends on the location and size of the circle.
Mike W.
(published on 08/22/2008)
(published on 05/16/2013)
(published on 09/17/2009)