The other day, I found myself with an interesting problem of approximating a circle with the enclosing square which seems to prove

Here is the offending paradox:

This is an example of how counterintuitive questions can be answered with a little calculus.

The key is to realize that no matter how closely we approximate the circle, the

Note carefully that as you get closer to 90 degrees,

If we take a quadrant of the circle - let's say the top left quadrant, moving counter clockwise from top to left - we can imagine that each inf…

**pi = 4.***The paradox was forwarded by a most interesting puzzle collector, Surajit Basu, a friend and life long inspiration. See Sonata for Unaccompanied Tortoise for why!*Here is the offending paradox:

This is an example of how counterintuitive questions can be answered with a little calculus.

The key is to realize that no matter how closely we approximate the circle, the

*orthogonal lines of the approximation formed by inverting the square corners*will never actually be tangential to the circle.Note carefully that as you get closer to 90 degrees,

*the horizontal line is much longer than the vertical. Same goes with the approximation at 0 and 180 - the vertical line is much larger than the horizontal component.*If we take a quadrant of the circle - let's say the top left quadrant, moving counter clockwise from top to left - we can imagine that each inf…