If they are equal, then yes, the area of the star equals the area of the circle or radius $i$.

$\displaystyle \pi i^2 = 4-2\sqrt{2} \rightarrow i^2 = \frac{4-2\sqrt{2}}{\pi}$

$\displaystyle i~ sin G = g~ sin I \rightarrow sin^2 G = \frac{g^2}{i^2} sin^2 I = \left( \frac{\pi}{4-2\sqrt{2}} \right) \left( \frac{1}{2} \right) sin^2 \frac{\pi}{4} = \frac{\pi}{16-8\sqrt{2}} $

$\displaystyle \rightarrow G = arcsin \left( \sqrt{\frac{\pi}{16-8\sqrt{2}}} \right) \approx 2.1823327 ~rad \approx 125.03845Â° $ (using $\pi - Arcsin(â€¦)$ to account for obtuse angle)

The shaded areas had very close to the same area, but not the same. Thus, the area of the circle and the area of the star were different by a little under 1 %, causing the error when you tried to back out pi.

*Apologies to my pre-algebra teacher for not putting my radicals back in the numerator.