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Let A be area of a regular n-gon with perimeter p.


Let r be the radius of the incircle (apothem) and R the radius of the circumcircle,

so that A-= pi r2 and A+ = pi R2 are their areas.


From a triangle dissection, A = rp/2; so p = 2A/r.


Let A be constant; show that as n increases, p decreases.

Write A = pi r2.
We know that A- < A < A+, so r < r < R.

As n increases, A- and A+ converge, monotonically, to A; A- from below, and A+ from above.
It follows that r and R converge, monotonically, to r; r from below, and R from above.

Thus as n (and r) increase, p = 2A/r decreases, monotonically, (to 2 pi r.)

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