Draw \(HI,GK\perp EF\)

Each angle of the octagon is : \(\dfrac{180\left(8-2\right)}{8}=135^0\)

\(\Rightarrow\widehat{HAF}=\widehat{GFA}=180^0-135^0=45^0\)

Let a be the side length of the octagon, then \(HI=IA=GK=KF=\dfrac{a}{\sqrt{2}};IK=HG=a\)

\(\Rightarrow AF=a\left(1+\sqrt{2}\right)\Rightarrow S_{AHF}=\dfrac{a\left(1+\sqrt{2}\right).\dfrac{a}{\sqrt{2}}}{2}=\dfrac{a^2\left(2+\sqrt{2}\right)}{4}\)

\(S_{ADF}=S_{ABCDEFGH}-S_{AFGH}-S_{ABCD}-S_{FED}\)

\(=S_{AFGH}+S_{ABEF}+S_{BCDE}-S_{AFGH}-S_{ABCD}-S_{FED}\)

\(=S_{ABEF}-S_{FED}=a.a\left(1+\sqrt{2}\right)-a.\dfrac{a}{\sqrt{2}}:2\)

\(=\dfrac{a^2\left(4+3\sqrt{2}\right)}{4}\)

The answer is : \(\dfrac{4+3\sqrt{2}}{4}:\dfrac{2+\sqrt{2}}{4}=1+\sqrt{2}\approx2.41\)