Question

一个正十二边形（ $12$ 条边）内接于半径为 $r$ 英寸的圆。以平方英寸为单位，该十二边形的面积为：

A regular dodecagon ( $12$ sides) is inscribed in a circle with radius $r$ inches. The area of the dodecagon, in square inches, is:

$3r^2$

$2r^2$

$\dfrac{3r^2\sqrt3}{4}$

$r^2\sqrt3$

$3r^2\sqrt3$

将十二边形分成 $12$ 个以圆心为一个顶点的三角形

Divide the dodecagon into $12$ triangles with vertex at the center

每个圆心角为 $30^\circ$ ，所以使用 $\frac12r^2\sin30^\circ$

Each central angle is $30^\circ$ , so use $\frac12r^2\sin30^\circ$

这 $12$ 个以圆心为顶点的三角形，每个面积为 $\frac12r^2\sin30^\circ=\frac{r^2}{4}\text{。}$ 因此总面积为 $12(\frac{r^2}{4})=3r^2$ 。

所以正确答案是 A。

Each of the $12$ central triangles has area $\frac12r^2\sin30^\circ=\frac{r^2}{4}.$ Their total area is $12(\frac{r^2}{4})=3r^2.$

Thus, the correct answer is A.