How do I get an integer from a polygon equation, which usually returns fractions?

Question

Known: edge length

Unknown: number of edges

Radius should increase of decrease to an interval to ensure number of edges in Polygon is divisible by 1

E.g. edge length is 100mm, number of edges unknown, radius is 6m. How do I find the closest radius to 6m which gives an integer, divisible by 1, for a realistic number of edges.

Previous Question

If a regular polygon has a fixed edge length, can I know how many edges it has by knowing the length from corner to its center?

& it's answer

the radius of the polygon, and it has the formula $$r=\frac{s}{2\sin\left(\frac{180°}{n}\right)}$$ where $s$ is the side length of the polygon and $n$ is the number of sides. So given $r$ and $s$, you can simply solve the above equation for $n$.

Asterix

It's worth pointing out that when you solve for n there's no guarantee that it will turn out to be an integer, and hence correspond to a regular polygon

I have no math background, not even enough to know which tags to attach. How do I solve for these intervals of radii?

Thanks for your help! :~)

Answer 1

Rearrange your formula for $r$ in terms of $n$

$$n = \frac \pi {\arcsin\left( \frac s {2r} \right)} \approx \frac {2 \pi r} s \quad \text{for } r \gg s$$

and insert the desired values. For your example $n \approx 2 \pi \frac{6 \text{ m}}{0.1 \text{ m}} \approx 377$ is very close

$$\begin{array}{c|c|c|c} n & 376 & 377 & 378 \\ \hline r \text{ in m} & 5.9843 & 6.0002 & 6.0161 \end{array}$$