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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.....

 Aug 10, 2023
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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

 

cool cool cool

 Aug 10, 2023

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