Each corner cube is removed from this 3text cm× 3text cm× 3text cm cube. The surface area of the remaining figure is

AMC 8 1997 Problem 21

AMC8 1997 年第 21 题

立体几何★★★☆☆

Each corner cube is removed from this

3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}

cube. The surface area of the remaining figure is

从一个

3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}

的正方体上移除每个角上的小正方体。剩余图形的表面积是

正确答案 · Correct Answer

Solution 1

The original cube has 6 square surfaces that each have an area of

3^2 = 9

, for a toal surface area of

6\cdot 9 = 54

Since no two corner cubes touch, we can examine the effect of removing each corner cube individually.

Each corner cube contribues 3 faces each of surface area 1 to the big cube, so the surface area is decreased by 3 when the cube is removed.

However, when the cube is removed, 3 faces on the 3x3x3 cube will be revealed, increasing the surface area by 3.

Thus, the surface area does not change with the removal of a corner cube, and it remains 54, which is answer

D

继续练习