3. Given that the equilateral $\triangle A B C$ is inscribed in $\odot O, S_{\triangle B B C}=a$, and the quadrilateral $D E F G$ is an inscribed square in the semicircle $\odot O$, $S_{\text {quadrilateral } D E F G}=b$. Then the value of $\frac{a}{b}$ is $($.
(A) 2
(B) $\frac{\sqrt{6}}{2}$
(C) $\frac{3 \sqrt{3}}{5}$
(D) $\frac{15 \sqrt{3}}{16}$
3. D.

Let the radius of $\odot O$ be $r$. Then
$$
a=\frac{3 \sqrt{3} r^{2}}{4}, b=\frac{4 r^{2}}{5} .
$$