Question

连接一个单位立方体相邻面的中心，形成一个正八面体。这个八面体的体积是多少？

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?

$\dfrac{1}{8}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

答案：B

知识点：立体几何棱锥体积

难度评级：1760

解答：

六个面中心形成一个正八面体，可看成两个全等的正方形棱锥共用底面。相邻面中心距离为 $\frac{\sqrt2}{2}$ ，所以正方形底面面积为 $\left(\frac{\sqrt2}{2}\right)^2 = \frac12$ 。

每个棱锥高为 $\frac12$ ，体积为 $\frac13 \cdot \frac12 \cdot \frac12 = \frac{1}{12}$ 。八面体体积为 $2 \cdot \frac{1}{12} = \frac16$ 。

所以正确答案是 B。

The six face centers form a regular octahedron, viewed as two congruent square pyramids sharing a base. Adjacent face centers are $\frac{\sqrt2}{2}$ apart, so the square base has area $\left(\frac{\sqrt2}{2}\right)^2 = \frac12.$

Each pyramid has height $\frac12,$ so its volume is $\frac13 \cdot \frac12 \cdot \frac12 = \frac{1}{12}.$ The octahedron has volume $2 \cdot \frac{1}{12} = \frac16.$

Thus, the correct answer is B.

2006 AMC 10A 第 24 题

答案：B

难度评级：1760

解答：

其他年份的第 24 题