Problem:
A sphere is inscribed in a cube that has a surface area of square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since the surface area of the original cube is square meters, each face of the cube has a surface area of square meters, and the side length of this cube is meters. The sphere inscribed within the cube has diameter meters, which is also the length of the diagonal of the cube inscribed in the sphere. Let represent the side length of the inscribed cube. Applying the Pythagorean Theorem twice gives
Hence each face has surface area
So the surface area of the inscribed cube is square meters.
The problems on this page are the property of the MAA's American Mathematics Competitions