Question

下图显示一个正方形和四个等边三角形。每个三角形都有一条边在正方形的一条边上，每个三角形边长为 $2$ ，且四个三角形的第三个顶点在正方形中心相交。正方形内但三角形外的区域被涂色。涂色区域的面积是多少？

The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length $2$ and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?

$4$

$12 - 4\sqrt{3}$

$3\sqrt{3}$

$4\sqrt{3}$

$16 - 4\sqrt{3}$

答案：B

知识点：面积分割等边三角形特殊直角三角形

难度评级：1330

解答：

三角形的高为 $2$ ，这是利用 $\sqrt3$ 三角形得到的，所以总底长为 $2\sqrt3$ 。每侧未被白色区域覆盖的总长度为 $12$ ，因此每个小三角形对应长度为。

因此共有 $\frac{\sqrt3}{4}(2^2)=\sqrt3$ 个小三角形，底为，高为，总面积为 $12-4\sqrt3.$

所以正确答案是 B。

Each equilateral triangle has side length $2$ , so its altitude is $\sqrt3$ . Because that altitude runs from a side of the square to its center, the square has side length $2\sqrt3$ and area $12$ .

Each of the four equilateral triangles has area $\frac{\sqrt3}{4}(2^2)=\sqrt3$ . Therefore the shaded area is $12-4\sqrt3.$

Thus, the answer is B .

2019 AMC 10B 第 8 题

答案：B

难度评级：1330

解答：

其他年份的第 8 题