BMAD Posted January 14, 2015 Report Share Posted January 14, 2015 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. Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted January 15, 2015 Report Share Posted January 15, 2015 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 r^{2} and A^{+} = pi R^{2} 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 r^{2}. 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.) Quote Link to comment Share on other sites More sharing options...

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## 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.

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