2013 AMC 10A 第 14 题

Question

从边长为

3

的实心立方体的每个角上各去掉一个边长为

1

的实心小立方体。剩下的立体有多少条棱？

A solid cube of side length

1

is removed from each corner of a solid cube of side length

3.

How many edges does the remaining solid have?

36

60

72

84

108

每去掉一个角，都会产生新的外露棱。

Each corner removal creates new exposed edges

数原立方体的棱，再加上每个被去掉角产生的新棱。

Count the original cube edges plus the new edges from each removed corner

去掉角上的小立方体不会消除原来大立方体的十二条外边界棱，只会增加新的棱。

每去掉一个角，会新增

9

条棱。

8

个角，所以新增

8 \cdot 9 = 72

条棱。再加上原来的

12

条棱，总数为

72 + 12 = 84

。

所以正确答案是 D。

Removing the cubes does not remove any edges from the original cube. It only adds edges.

After removing each cube, we can see that

9

extra edges are added to the solid.

8

cubes are removed, which means

8 \cdot 9 = 72

edges are added to the original

12

edges, for a total of

72 + 12 = 84

edges.

Thus, D is the correct answer.

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