19.38 A regular $n$-sided polygon is inscribed in a circle of radius $R$, and the area of this $n$-sided polygon is $3 R^{2}$. Then $n$ equals
(A) 8.
(B) 10.
(C) 12.
(D) 15.
(E) 18.
(12th American High School Mathematics Examination, 1961)
[Solution] Given $S=\frac{1}{2} p r$, where $p$ is the perimeter of a regular polygon, $r$ is the apothem, and let the side length be $a_{n}$.
Also, $p=n a_{n}=n \cdot 2 R \sin \frac{180^{\circ}}{n}, r=R \cos \frac{180^{\circ}}{n}$.
Given $3 R^{2}=\frac{1}{2} \cdot n \cdot 2 R \sin \frac{180^{\circ}}{n} \cdot R \cos \frac{180^{\circ}}{n}$,

which simplifies to
$$
\frac{6}{n}=\sin \frac{360^{\circ}}{n} .
$$

For $n \geqslant 3$ as an integer, the only value that satisfies this equation is $n=12$.
Therefore, the answer is $(C)$.