To find the surface area of the largest sphere that can fit inside a cube box, we first need to determine the radius of the sphere.
Given that the box is a cube with dimensions of 20 cm × 20 cm × 20 cm, the largest sphere that can fit inside will have a diameter equal to the side length of the cube. Therefore, the diameter of the sphere is 20 cm.
The radius \( r \) of the sphere is half of the diameter:
\[ r = \frac{20 , \text{cm}}{2} = 10 , \text{cm} \]
The formula for the surface area \( A \) of a sphere is given by:
\[ A = 4\pi r^2 \]
Substituting the radius into the formula:
\[ A = 4\pi (10 , \text{cm})^2 \] \[ A = 4\pi (100 , \text{cm}^2) \] \[ A = 400\pi , \text{cm}^2 \]
Thus, the surface area of the largest sphere that can fit in the box is:
\[ \boxed{400\pi} , \text{cm}^2 \]