23. Given that the sides $A B$ and $A C$ of the equilateral $\triangle A B C$ are tangents to the circle $\odot O$, with points of tangency at $B$ and $C$ respectively. Then the ratio of the area of the part of $\triangle A B C$ outside the circle to the area of the triangle is ( ).
(A) $\frac{4 \sqrt{3} \pi}{27}-\frac{1}{3}$
(B) $\frac{\sqrt{3}}{2}-\frac{\pi}{8}$
(C) $\frac{1}{2}$
(D) $\sqrt{3}-\frac{2 \sqrt{3} \pi}{9}$
(E) $\frac{4}{3}-\frac{4 \sqrt{3} \pi}{27}$
23. E.

Let the radius of $\odot O$ be $r$. Then
$$
\frac{r \times \sqrt{3} r - \frac{1}{3} \pi r^{2}}{\frac{\sqrt{3}}{4}(\sqrt{3} r)^{2}} = \frac{4}{3} - \frac{4 \sqrt{3} \pi}{27} .
$$