Let $P$ be a regular nonagon with side length $2,$ and let $S$ be the set of points that are within a distance of $1$ from $P.$ (That is, a point $X$ is in $S$ if there exists a point $Y$ in $P$ such that $XY \le 1.$) What is the perimeter of $S?$

Apparently, the answer has $\pi$ in it. Also, someone told me this has to do with funky circle areas.....

Guest Aug 10, 2023

#1**+1 **

I'm unsure if S is supposed to lie outside or inside of P.....here, I've assumed outside

Point FC = 1.....so AF / sin 70° = FC / sin 20° so that AF = (sin 70°)/(sin20°)

F = (0, sin 70°/sin 20°)

DF = 1

So....the radius of S = (1 + sin 70°/sin20°)

The exact perimeter = 2 ( 1 + sin70°/sin20°) pi = ( 2 + 2sin 70°/sin20°) pi ≈ 7.5 pi

CPhill Aug 10, 2023