Shrinking Sides Problem

Geometry Level 3

Consider the following sequence of regular polygons: You start with an equilateral triangle with side length 1, then go to a square with sides of 1/2, a pentagon with sides of 1/3, a hexagon with sides of 1/4, and so on, where the side lengths of your n-gon are: 1/(n -2), where n is the number of sides. Keep in mind that all polygons must be regular. Does the sum of the areas of this sequence of shapes converge or diverge? Also, I think it might be fun to think up more problems like this (with a sequence of shapes with a different pattern of changing side lengths and their areas) and solve them.

Converges Diverges

1 solution

Mark Hennings
Nov 3, 2019

Since the area of a regular $n$ -gon with side length $r$ is $\tfrac14nr^2\cot\tfrac{\pi}{n}$ , the area of a regular $n$ -gon with side length $(n-2)^{-1}$ is $A_n = \frac14\times \frac{n}{(n-2)^2}\cot\tfrac{\pi}{n} \; = \; \frac{1}{4\pi}\left(\frac{n}{n-2}\right)^2 \tfrac{\pi}{n} \cot\tfrac{\pi}{n} \to \frac{1}{4\pi} \hspace{2cm} n \to \infty$ so the series $\sum_{n \ge 3}A_n$ must diverge, since $A_n$ does not converge to $0$ .

Shrinking Sides Problem

1 solution

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