Question

一个单位立方体被切两刀，形成三个三棱柱，其中两个全等，如图一所示。然后按图二中虚线所示用同样方式切割该立方体，得到九块。包含顶点 $W$ 的那一块体积是多少？

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W?$

$\dfrac{1}{12}$

$\dfrac{1}{9}$

$\dfrac{1}{8}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

答案：A

知识点：棱锥体积立体几何

难度评级：1910

解答：

两组切割都从上边沿切到下底面的中线附近。在 $W$ 附近，它们切出一个棱锥，其顶点是 $W$ 正上方的顶点。

它的底面是边长 $\dfrac{1}{2}$ 的正方形，即下底面的四分之一，高为 $1$ 。因此体积为 $\dfrac{1}{3} \left(\dfrac{1}{2}\right)^2 (1) = \dfrac{1}{12}.$

所以正确答案是 A。

The two perpendicular sets of cuts each run from top edges to midlines of the bottom face. Near $W$ they carve out a square pyramid. Its base is the quarter of the bottom face adjacent to $W,$ and its apex lies one unit above the center of the bottom face.

Thus its base is a square of side $\dfrac{1}{2}$ and its altitude is the full height $1.$ Therefore the volume is $\dfrac{1}{3} \left(\dfrac{1}{2}\right)^2 (1) = \dfrac{1}{12}.$

Thus, the correct answer is A.

2005 AMC 12A 第 17 题

答案：A

难度评级：1910

解答：

其他年份的第 17 题