AMC 8 1997 Problem 17
AMC8 1997 年第 17 题
立体几何★★★☆☆
A cube has eight vertices (corners) and twelve edges. A segment, such as , which joins two vertices not joined by an edge is called a diagonal. Segment is also a diagonal. How many diagonals does a cube have?
一个正方体有八个顶点和十二条棱。像 这样连接两个不被棱相连的顶点的线段称为对角线。线段 也是一条对角线。一个正方体有多少条对角线？
- A.6
- B.8
- C.12
- D.14
- E.16
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
On each face, there are 2 diagonals like . There are 6 faces on a cube. Thus, there are 2×6 = 12 diagonals that are "x-like".
Every "y-like" diagonal must connect the bottom of the cube to the top of the cube. Thus, for each of the 4 bottom vertices of the cube, there is a different "y-like" diagonal. So there are 4 "y-like" diagonals.
This gives a total of 12 + 4 = 16 diagonals on the cube, which is answer .
There are 8 vertices on a cube(A, B, C, D, E, F, G, H). If you pick one vertice, it will have 7 line segments associated with it, so you will have segments within the cube. The division by 2 is necessary because you counted both the segment from to and the segment from to .
But not all 28 of these segments are diagonals. Some are edges. There are 4 edges on the top, 4 edges on the bottom, and 4 edges that connect the top to the bottom. So there are 12 edges total, meaning that there are 28 - 12 = 16 segments that are not edges. All of these segments are diagonals, and thus the answer is .
Consider picking one point on the corner of the cube. That point has 3 "x-like" diagonals, and 1 "y-like" diagonal that ends on the opposite vertex. Thus, each vertex has 3 + 1 = 4 diagonals associated with it. There are 8 vertices on the cube, giving a total of diagonals.
However, each diagonal was counted as both a "starting point" and an "ending point". So there are really diagonals, giving an answer of .
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