Question

如下图所示，把长方体各个面的中心连接起来，形成一个八面体。这个八面体的体积是多少？

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of this octahedron?

$\dfrac{75}{12}$

$10$

$12$

$10\sqrt{2}$

$15$

答案：B

知识点：多面体棱锥体积

难度评级：1420

解答：

这个八面体可看成两个全等棱锥，它们共用的底面是通过四个侧面中心的菱形。该菱形对角线为 $4$ 和 $5$ ，面积为 $\frac12\cdot4\cdot5=10$ 。

两个棱锥的高都为 $\frac32$ ，所以八面体体积为 $2\cdot\frac13\cdot10\cdot\frac32=10.$

所以正确答案是 B。

The octahedron can be viewed as two congruent pyramids whose shared base is the rhombus through the centers of the four side faces. This rhombus has diagonals $4$ and $5$ , so its area is $\frac12\cdot4\cdot5=10$ .

Each pyramid has height $\frac32$ , half the prism's height. Thus the total volume is $2\cdot\frac13\cdot10\cdot\frac32=10.$

Thus, the correct answer is B.

2015 AMC 10B 第 17 题

答案：B

难度评级：1420

解答：

其他年份的第 17 题