Circumference inscribed in a square What is the area of the hatched region, knowing that the arc AC is 1/4 of the circumference with a center in D?

I've tried using algebra to solve this but it seemed insufficient, I thought of using integrals to find the area, by finding the analytic geometry equations for the circles but it went nowhere.


Sector 1 Angle $= 2\cdot \sin^{-1}(\sqrt(\frac{31}{128})\cdot 2)= 2.78617$ rad
Sector 2 Angle $= 2\cdot \sin^{-1}(\sqrt(\frac{31}{128})) = 1.02906$ rad
Shaded Area = Area Chord Segment 1 – Area Chord Segment 2
Area Seg 1 = Area Sector 1  - Area Triangle 1
$\frac{2.78617}{2π}\cdot π(\frac{a}{2})^2 - (\sqrt(\frac{31}{128})a)( \sqrt(\frac{97}{128})a - \sqrt(2)\frac{a}{2})$
                $= .26785 a^2$
A Seg 2 = Area Sector 2 - Area Triangle 2
$\frac{1.02906}{2π}\cdot πa^2 – (\sqrt(\frac{31}{128})a(\sqrt(\frac{97}{128})a)$
                $= .08612 a^2$
Shaded  Area $= (.26785 - .08612) a^2 =  .18173 a^2$
