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Question Number 186052 by Rupesh123 last updated on 31/Jan/23

	Answered by HeferH last updated on 31/Jan/23

	Commented by HeferH last updated on 31/Jan/23

	$S : a r e a o f r e g u l a r o c t a g o n$ $S_{1} : a r e a s h a d e d$ $r : i n r a d i u s$ $S = 4 [\frac{(r - \frac{r}{\sqrt{2}}) r \sqrt{2}}{2}] + {(r \sqrt{2})}^{2} = \frac{r^{2} (3 + \sqrt{2})}{2}$ $s p e c i a l t r i a n g l e A B C (s i d e s : k, k \sqrt{2}, k \sqrt{3}) \Rightarrow$ $B C = D C = \frac{r}{\sqrt{2}} \Rightarrow A B = r$ $\Rightarrow S_{1} = \frac{r^{2}}{2 \sqrt{2}}$ $\frac{S_{1}}{S} = \frac{r^{2}}{2 \sqrt{2}} \cdot \frac{2}{r^{2} (3 + \sqrt{2})} = \frac{1}{3 \sqrt{2} + 2} = \frac{3 \sqrt{2} - 2}{14}$
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