Octagon circumscribing a circle

The side length of the octagon is $l=\frac{96}{24}=4$ . There are many ways to find the radius of the inscribed circle, the direct formula uses trigonometry, but there is another way. First we find the radius of the circumscribed circle, then we find the radius of the inscribed circle (the length of the apothem).

The radius $R$ of the circumscribed circle in an octagon can be found as following:

$\displaystyle{l=R\sqrt{2-\sqrt{2}}}$

$\displaystyle{4=R\sqrt{2-\sqrt{2}}}$

$\displaystyle{R=\frac{4}{\sqrt{2-\sqrt{2}}}}$

The length of the inscribed circle radius $r$ can be found as following:

$\displaystyle{r=\frac{R}{2}\sqrt{2+\sqrt{2}}}$

$\displaystyle{r=\frac{\frac{4}{\sqrt{2-\sqrt{2}}}}{2} \sqrt{2+\sqrt{2}}}$

$\displaystyle{r=\frac{4\sqrt{2+\sqrt{2}}}{2\sqrt{2-\sqrt{2}}}}$

$\displaystyle{r=\frac{4\sqrt{2+\sqrt{2}}\sqrt{2-\sqrt{2}}}{2\sqrt{2-\sqrt{2}}\sqrt{2-\sqrt{2}}}}$

$\displaystyle{r=\frac{4\sqrt{4-2}}{2(2-\sqrt{2})}}$

$\displaystyle{r=\frac{2\sqrt{2}}{2-\sqrt{2}}}$

Then the area of the inscribed circle will be:

$\displaystyle{\left(\frac{2\sqrt{2}}{2-\sqrt{2}}\right)^2 \cdot \pi \approx \boxed{73}}$