20.93 A circle is inscribed in an equilateral triangle, and a square is inscribed in this circle. Then the ratio of the area of the triangle to the area of the square is
(A) $\sqrt{3}: 1$.
(B) $\sqrt{3}: \sqrt{2}$.
(C) $3 \sqrt{3}: 2$
(D) $3: \sqrt{2}$.
(E) $3: 2 \sqrt{2}$.
(14th American High School Mathematics Examination, 1963)
[Solution] Let the radius of the circle be $r$, and the side lengths of the equilateral triangle and the square be $a$ and $b$, respectively. Then $a=2 \sqrt{3} r, \quad b=\sqrt{2} r$.
$S_{\triangle}=\frac{\sqrt{3}}{4} a^{2}=3 \sqrt{3} r^{2}$,
$S_{\square}=b^{2}=2 r^{2}$.

Therefore, $S_{\triangle}: S_{\square}=3 \sqrt{3}: 2$.
Hence, the answer is $(C)$.