It's All N-Gons.

Level pending

The above diagram represents a regular $n - gon$ , where $n$ is an even integer and $n \geq 4$ .

(1) Using the above diagram find the length $x$ of a side of the $n - gon$ .

(2) Using $n = 100$ , find the length $x$ of a side to eight decimal places.

1 solution

Rocco Dalto
Nov 27, 2019

Let $|\overline {\rm AB}| = 2r$ and $n$ be even integer and $n \geq 4$ .

$x = 2r\sin(\dfrac{\pi}{n})$ and using right $\triangle{ABC}$ in the first diagram above we have:

$(2r)^2 = 21^2 + 3^2 = 450 \implies r = \dfrac{15\sqrt{2}}{2} = \dfrac{15}{\sqrt{2}} \implies$

$x = 15\sqrt{2}\sin(\dfrac{\pi}{n})$ .

Using $n = 100 \implies x = 15\sqrt{2}\sin(\dfrac{\pi}{100}) \approx \boxed{0.66632282}$ .