CAT 2016 Quantitative Ability

Let the length of the side of the octagon and the radius of the circle be ‘a’ units and T’ units respectively. Therefore, the length of the side of the square will be $\left(\frac{2\mathit{r}}{\sqrt{2}}\right)$ units.LM= LA+AB+BM
⇒ $2\mathit{r}=\frac{\mathit{a}}{\sqrt{2}}+\mathit{a}+\frac{\mathit{a}}{\sqrt{2}}$
⇒ $\mathit{r}=\frac{\mathit{a}\left(1+\sqrt{2}\right)}{2}$
Hence, the ratio
= $\frac{{\left(\frac{2\mathit{r}}{\sqrt{2}}\right)}^2}{8\times \frac{1}{2}\times \mathit{r}\times \mathit{a}}$
$=\frac{\mathit{r}}{2\mathit{a}}$
$=\frac{\mathit{a}\left(\frac{1+\sqrt{2}}{2}\right)}{2\mathit{a}}$
$=\left(1+\sqrt{2}\right):4$