+107 A regular octagon withside length 4 cm is concentricwith a circle inside, If the area of the circle is equal to the area to the shaded region between the shapes, what is the radius of the circle?
+26367 A regular octagon withside length 4 cm is concentricwith a circle inside,
If the area of the circle is equal to the area to the shaded region between the shapes,
what is the radius of the circle?
\(\text{Let $s =$ octagon side length $ = 4~ cm$}\\ \text{Let $a =$ octagon apothem length}\)
\(\begin{array}{|rcll|} \hline A_{\bigcirc} &=& A_{\text{octagon}} - A_{\bigcirc} \\ 2A_{\bigcirc} &=& A_{\text{octagon}} \\ \mathbf{A_{\bigcirc}} &=& \mathbf{\dfrac{A_{\text{octagon}}} {2}} \\ \hline \end{array}\)
apothem
\(\begin{array}{|rcll|} \hline a &=& \dfrac{s}{2}\cdot \tan\left(90^{\circ}-\frac{45^{\circ}}{2} \right) \\ a &=& \dfrac{s}{2}\cdot \left(1+\sqrt{2} \right) \\\\ A_{\text{octagon}} &=& \dfrac{a \cdot s}{2} \cdot 8 \\ &=& 4as \\ &=& 4 \left( \dfrac{s}{2}\cdot \left(1+\sqrt{2} \right) \right)s \\ &=& 2 s^2 \left(1+\sqrt{2} \right) \\ \mathbf{ \dfrac{A_{\text{octagon}}} {2} } & \mathbf{=} & \mathbf{s^2 \left(1+\sqrt{2} \right)} \\ \hline \end{array} \)
\(\text{Let $A_{\bigcirc} = \pi r^2$ }\)
\(\begin{array}{|rcll|} \hline \mathbf{A_{\bigcirc}} &=& \mathbf{\dfrac{A_{\text{octagon}}} {2}} \\ \pi r^2 &=& s^2 \left(1+\sqrt{2} \right) \\\\ r^2 &=& \dfrac{s^2 \left(1+\sqrt{2} \right)} {\pi} \\\\ r &=& s \sqrt{ \dfrac{ 1+\sqrt{2} } {\pi} } \quad & | \quad s=4~ cm\\\\ r &=& 4 \sqrt{ \dfrac{ 1+\sqrt{2} } {\pi} } \\\\ r &=& 4 \cdot 0.87662309134 \\ r &=& 3.50649236534 \\ \mathbf{r} & \mathbf{=} & \mathbf{3.5~ cm} \\ \hline \end{array}\)
