Common area = Area of Δ ACD - 2 x Area of Δ CDE
= \(\frac{a^2}2-2\times\frac12\times cd^2\)
= \(\frac{a^2}2-(OC-OD)^2\)
= \(\frac{a^2}2-(\frac a{\sqrt2}-\frac a2)^2\)
= \(\frac{a^2}2-\frac{a^2}2(1-\frac 1{\sqrt2})^2\)
= \(\frac{a^2}2-(1-(1+\frac12-\frac2{\sqrt2}))\)
= \(\frac{a^2}2-(\sqrt2-\frac12)\)
= \(\frac{a^2}4(2\sqrt2-1)\) square units.