AMC12 2005 A

AMC12 2005 A · Q17

AMC12 2005 A · Q17. It mainly tests 3D geometry (volume).

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 $W$?

一个单位立方体被切割两次，形成三个三角柱体，其中两个全等，如图 1 所示。然后按照图 2 中虚线所示的方式，用同样的方法再次切割该立方体。这将产生九个部分。包含顶点 $W$ 的那一块的体积是多少？

(A) \frac{1}{12} \frac{1}{12}

(B) \frac{1}{9} \frac{1}{9}

(D) \frac{1}{6} \frac{1}{6}

(E) \frac{1}{4} \frac{1}{4}

Answer

Correct choice: (A)

正确答案：（A）

Solution

It is a pyramid with height $1$ and base area $\frac{1}{4}$, so using the formula for the volume of a pyramid, $\frac{1}{3} \cdot \left(\frac{1}{4}\right) \cdot (1) = \frac {1}{12} \Rightarrow \boxed{(\mathrm {A})}$.

它是一个高为 $1$、底面积为 $\frac{1}{4}$ 的棱锥，因此用棱锥体积公式，$\frac{1}{3} \cdot \left(\frac{1}{4}\right) \cdot (1) = \frac {1}{12} \Rightarrow \boxed{(\mathrm {A})}$.

Topics

GE.013 3D geometry (volume)