P(A)

[BMAD]

find a formula for the perimeter of a regular polygon as a function of its area. It needs to be shown that as the number of sides of a polygon of given area increases, the perimeter of the figure

decreases.

[bonanova]

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.)