1989 AMC 12 第 26 题
先试着解答 1989 AMC 12 第 26 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 1989 AMC 12 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
26.
连接一个立方体相邻各面的中心，形成一个正八面体。该八面体与立方体的体积之比为
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
小提示：
将立方体的边长缩放为 ，并把它的中心置于原点
Scale the cube to side length and place its center at the origin
大提示：
该八面体的顶点为
The octahedron has vertices
解答：
取边长为 的立方体，则其体积为 。各面的中心为 。在每个卦限中，八面体截出一个三条互相垂直的棱长均为单位长度的四面体，其体积为 。所以八面体的体积为 ，所求体积比为 。
所以正确答案是 C。
Take a cube of side so its volume is The face centers are In each octant, the octahedron cuts out a tetrahedron with three perpendicular unit edges and volume Thus its volume is and the ratio is
Thus the correct answer is C.
其他年份的第 26 题
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