Question

一个球内切于表面积为 $24$ 平方米的立方体。再将第二个立方体内接于这个球。内部立方体的表面积是多少平方米？

A sphere is inscribed in a cube that has a surface area of $24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?

$3$

$6$

$8$

$9$

$12$

答案：C

知识点：正方体球表面积

难度评级：1580

解答：

外部立方体每个面的面积为 $24 \div 6 = 4$ ，所以边长为 $2$ ，球的直径为 $2$ 。

这个直径是内部立方体的空间对角线，所以 $l\sqrt3 = 2$ ，得到 $l^2 = \tfrac43$ 。

内部立方体表面积为 $6 l^2 = 6 \cdot \tfrac43 = 8$ 。

所以正确答案是 C。

Each face of the outer cube has area $24 \div 6 = 4,$ so its side is $2,$ and the sphere has diameter $2.$

This diameter is the space diagonal of the inner cube, so $l\sqrt3 = 2,$ giving $l^2 = \tfrac43.$

The inner cube's surface area is $6 l^2 = 6 \cdot \tfrac43 = 8.$

Thus, the correct answer is C.

2007 AMC 10A 第 21 题

答案：C

难度评级：1580

解答：

其他年份的第 21 题