A circle, a square and an octangle walk into a bar
A circle is inscribed into a square of side \( a = 13 \). The circle is also inscribed into an octangle. Exactly four octangle's sides are parallel with square's sides, and other four are perpendicular with appropriate square's diagonals, as shown in the image above.
Find the area of the green region.
Give your answer to 3 decimal places.