This figure is a regular polygon of 8 sides.
The first step is to find the interior angle of this polygon, using the formula:
[tex]a_i=\frac{(n-2)\cdot180}{n}_{}[/tex]
Where n is the number of vertices of the polygon. So, using n = 8, we have:
[tex]a_i=\frac{6\cdot180}{8}=135\degree[/tex]
So the interior angle of a regular octagon is 135°. The segment that connects one vertex and the center of the figure divide these interior angles in 2 equal angles, so we have small isosceles triangles with base angle equal to 67.5°.
In order to find the base and height of this small triangle, we can do the following:
[tex]\begin{gathered} \cos (67.5\degree)=\frac{\frac{b}{2}}{9} \\ 0.38268=\frac{b}{18} \\ b=6.888 \\ \\ \sin (67.5\degree)=\frac{h}{9} \\ \text{0}.92388=\frac{h}{9} \\ h=8.135 \end{gathered}[/tex]
The perimeter of the octagon is:
[tex]P=8b=8\cdot6.888=55.104[/tex]
And the area is:
[tex]\begin{gathered} A=8\cdot\frac{b\cdot h}{2} \\ A=4\cdot6.888\cdot8.135 \\ A=224.136 \end{gathered}[/tex]
