Polka Dots in Pi

Geometry Level 4

In the figure, \(ABCD\) is a square of side length \(a\) units and \(E,F,G,H\) are the midpoints of the sides.

\(EAH, FBE,GCF,HDG\) are quadrants. There are two semicicles with diameters \(AD\) and \(BC\).

If the area of the shaded region in red can be expressed in the form

\[ \dfrac{a^2}x \left( \sqrt y - \dfrac\pi z\right) \]

then submit \( \dfrac {xy} z\) as your answer.


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