Question

如下面第一幅图所示，连接两个相邻边的中点，将三个 $12\text{ cm}\times12\text{ cm}$ 的正方形分别切成 $A$ 和 $B$ 两块。再如第二幅图所示，将这六块拼接到一个正六边形上，以便折成一个多面体。这个多面体的体积（单位： $\text{cm}^3$ ）是多少？

Three $12\text{ cm}\times12\text{ cm}$ squares are each cut into two pieces $A$ and $B,$ as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$ ) of this polyhedron?

答案

答案：864

知识点：正方体展开图（立体几何）体积

难度评级：2720

小提示：

看出每个 $A$ 是切去一个角的正方形面，而每个 $B$ 正是被切下的角三角形

Recognize each $A$ as a square face with one corner cut off and each $B$ as that corner triangle

大提示：

一个经过正方体六条棱中点的平面截出正六边形，并将正方体分成两个全等部分

A plane through six edge midpoints of a cube cuts a regular hexagon and divides the cube into two congruent parts

解答：

考虑一个 $12\times12\times12$ 的正方体，以及经过六条棱中点的平面；这六条棱连接两组互为相对的三个顶点。在坐标范围 $0\leq x,y,z\leq12$ 内，该平面为 $x+y+z=18$ 。它截出的截面是边长为 $6\sqrt2$ 的正六边形，其边长与连接正方形相邻边中点的切割线段相同。

在正方体的三个面上，该平面留下一个切去角三角形的正方形，恰好就是 $A$ 块。在另外三个面上，它留下互补的等腰直角三角形，恰好就是 $B$ 块。因此，图示展开图就是该平面切分正方体所得的两部分之一。中心对称会交换这两部分，所以每部分的体积都是正方体体积的一半： $V=\frac{12^3}{2}=864\text{。}$

Consider a $12\times12\times12$ cube and the plane through the midpoints of the six edges that join opposite groups of three vertices. In coordinates $0\leq x,y,z\leq12,$ this is the plane $x+y+z=18.$ Its cross-section is a regular hexagon with side $6\sqrt2,$ the same as the cut edge joining adjacent side midpoints.

On three faces of the cube, the plane leaves a square with a corner triangle removed, exactly piece $A.$ On the other three faces, it leaves the complementary right-isosceles corner triangle, exactly piece $B.$ Thus the pictured net is one of the two pieces into which this plane cuts the cube. Central symmetry interchanges the two pieces, so each has half the cube’s volume: $V=\frac{12^3}{2}=864.$

1985 AIME 第 15 题

答案：864

难度评级：2720

小提示：

大提示：

解答：

其他年份的第 15 题