Question

一个 $n$ 边正多边形内接于半径为 $R$ 的圆。多边形的面积为 $3R^2$ 。则 $n$ 等于：

A regular polygon of $n$ sides is inscribed in a circle of radius $R.$ The area of the polygon is $3R^2.$ Then $n$ equals:

$8$

$10$

$12$

$15$

$18$

将多边形分成 $n$ 个顶点在圆心的全等三角形

Divide the polygon into $n$ congruent triangles with vertex at the center

使用面积 $\frac12nR^2\sin(\frac{360^\circ}{n})$ 检验各选项

Use area $\frac12nR^2\sin(\frac{360^\circ}{n})$ and test the choices

多边形的面积为 $\frac n2R^2\sin\frac{360^\circ}{n}\text{。}$ 当 $n=12$ 时，它变为 $6R^2\sin30^\circ=3R^2\text{，}$ 符合要求。

因此，正确答案是 C。

The polygon’s area is $\frac n2R^2\sin\frac{360^\circ}{n}.$ For $n=12,$ this becomes $6R^2\sin30^\circ=3R^2,$ as required.

Therefore, the correct answer is C.