we can do a triangle to solve the apothem
the upper angle is equal to 360 divided by 8 since 360 is a complete turn and an octagon is composed of 8 equal triangles, so is 45
now we take a triangle from the triangle to apply trigonometric ratios and solve a
the upper angle of the new triangle is the half of the original so is 45/2
now i will use the trigonometric ratio of tangent
[tex]\tan (\alpha)=\frac{O}{A}[/tex]
where alpha is the angle, O the opposite side drom the angle and a the adjacent side from the angle
so replacing
[tex]\tan (\frac{45}{2})=\frac{3.5}{a}[/tex]
and solve for a
[tex]\begin{gathered} a=\frac{3.5}{\tan (\frac{45}{2})} \\ \\ a\approx8.4 \end{gathered}[/tex]
the aphotem is 8.4
and the formula of the area of the octagon is
[tex]A=4\times a\times l[/tex]
where a is the aphotem and l the measure f each side so 7
replacing
[tex]\begin{gathered} A=4\times8.4\times7 \\ A=235.2 \end{gathered}[/tex]
the area is 235.2 square units
