Question

一个边长为 $6$ 的正六边形，每条边外接一个等腰三角形。每个等腰三角形有两条边长为 $8$ 。将这些等腰三角形折起，形成一个以该六边形为底面的棱锥。这个棱锥的体积是多少？

A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8.$ The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?

$18$

$162$

$36\sqrt{21}$

$18\sqrt{138}$

$54\sqrt{21}$

答案：C

知识点：棱锥体积勾股定理

难度评级：1900

解答：

六边形中心到顶点的距离为 $6$ 。侧棱长为 $8$ ，所以棱锥高为 $\sqrt{8^2 - 6^2} = \sqrt{28} = 2\sqrt7$ 。

六边形面积为 $\dfrac{3\sqrt3}{2}\cdot 6^2 = 54\sqrt3$ 。因此体积为 $\dfrac13 \cdot 54\sqrt3 \cdot 2\sqrt7 = 36\sqrt{21}$ 。

因此，正确选项是 C。

The distance from the hexagon's center to a vertex is $6.$ A lateral edge has length $8,$ so the pyramid's height is $\sqrt{8^2 - 6^2} = \sqrt{28} = 2\sqrt7.$

The hexagon's area is $\dfrac{3\sqrt3}{2}\cdot 6^2 = 54\sqrt3.$ Thus the volume is $\dfrac13 \cdot 54\sqrt3 \cdot 2\sqrt7 = 36\sqrt{21}.$

Thus, the correct answer is C.

2015 AMC 12B 第 16 题

答案：C

难度评级：1900

解答：

其他年份的第 16 题