Question

点 $O$ 是正八边形 $ABCDEFGH$ 的中心， $X$ 是边 $\overline{AB}$ 的中点。阴影部分占八边形面积的几分之几？

Point $O$ is the center of the regular octagon $ABCDEFGH,$ and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded?

$\dfrac{11}{32}$

$\dfrac{3}{8}$

$\dfrac{13}{32}$

$\dfrac{7}{16}$

$\dfrac{15}{32}$

答案：D

知识点：正多边形面积分割

难度评级：1020

视频讲解：

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文字解答：

注意，将 $O$ 与各个顶点相连，可以把八边形分成 $8$ 个面积相等的三角形。因此每个三角形占八边形总面积的 $\dfrac {1}{8}$ 。

阴影部分包含三个完整三角形和三角形 $ABO$ 的一半。因此阴影面积占总面积的 $\dfrac{3.5}{8} = \dfrac{7}{16}$

所以正确答案是 D。

First notice that there are $8$ equally sized triangles that can be created with $O$ and any two consecutive points. Therefore, they each take up $\dfrac {1}{8}$ of the total area of the octagon.

The shaded area has three complete triangles and half of the triangle $ABO.$ Therefore, the shaded area is $\dfrac{3.5}{8} = \dfrac{7}{16}$ of the total area of the octagon.

Thus, the correct answer is D .

2015 AMC 8 第 2 题

答案：D

难度评级：1020

视频讲解：

文字解答：

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