A cube box is 20 cm×20 cm× 20 cm . What is the surface area of the largest size sphere that can fit in this box? Leave your answer in terms of pi.(
The largest sphere that can fit in a cube box touches all six faces of the cube at its points of tangency. Thus, the diameter of the sphere is equal to the length of the diagonal of the cube.
Using the Pythagorean theorem, the length of the diagonal of the cube is $\sqrt{20^2 + 20^2 + 20^2} = 20\sqrt{3}$.
Therefore, the diameter of the largest sphere is $20\sqrt{3}$, and the radius is $\frac{20\sqrt{3}}{2} = 10\sqrt{3}$.
The surface area of a sphere is given by the formula $4\pi r^2$. Therefore, the surface area of the largest sphere that can fit in the cube box is $4\pi (10\sqrt{3})^2 = \boxed{120\pi}$.