Question

What is the perimeter of a decagon with an apothem of 15 mm?

Answer 1

`\textit{apothem of a regular polygon}\\\\ a=\cfrac{ ~~ s\cot((180)/(n)) ~~ }{2} ~~ \begin{cases} n=\stackrel{number~of}{sides}\\ s=\stackrel{length~of}{a~side}\\[-0.5em] \hrulefill\\ n=10\\ a=15 \end{cases}\implies 15=\cfrac{ ~~ s\cot((180)/(10)) ~~ }{2} \\\\\\ 30=s\cot(18^o)\implies \cfrac{30}{\cot(18^o)}=s \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ \textit{sum of all ten sides or Perimeter} }{(10)\cfrac{30}{\cot(18^o)}\implies \cfrac{300}{\cot(18^o)} ~~ \approx ~~ \text{\LARGE 97.48}~mm}`

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