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# Octagon and Circle

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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?

Dec 18, 2018

#1
+22528
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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

$$\begin{array}{|rcll|} \hline a &=& \dfrac{s}{2}\cdot \tan\left(90^{\circ}-\frac{45^{\circ}}{2} \right) \\ a &=& \dfrac{s}{2}\cdot \left(1+\sqrt{2} \right) \\\\ A_{\text{octagon}} &=& \dfrac{a \cdot s}{2} \cdot 8 \\ &=& 4as \\ &=& 4 \left( \dfrac{s}{2}\cdot \left(1+\sqrt{2} \right) \right)s \\ &=& 2 s^2 \left(1+\sqrt{2} \right) \\ \mathbf{ \dfrac{A_{\text{octagon}}} {2} } & \mathbf{=} & \mathbf{s^2 \left(1+\sqrt{2} \right)} \\ \hline \end{array}$$

$$\text{Let A_{\bigcirc} = \pi r^2 }$$

$$\begin{array}{|rcll|} \hline \mathbf{A_{\bigcirc}} &=& \mathbf{\dfrac{A_{\text{octagon}}} {2}} \\ \pi r^2 &=& s^2 \left(1+\sqrt{2} \right) \\\\ r^2 &=& \dfrac{s^2 \left(1+\sqrt{2} \right)} {\pi} \\\\ r &=& s \sqrt{ \dfrac{ 1+\sqrt{2} } {\pi} } \quad & | \quad s=4~ cm\\\\ r &=& 4 \sqrt{ \dfrac{ 1+\sqrt{2} } {\pi} } \\\\ r &=& 4 \cdot 0.87662309134 \\ r &=& 3.50649236534 \\ \mathbf{r} & \mathbf{=} & \mathbf{3.5~ cm} \\ \hline \end{array}$$

Dec 18, 2018