To determine the largest sphere that can fit inside a rectangular box, we need to find the diameter of the sphere, which will be equal to the smallest dimension of the box.
The dimensions of the box are:
- Length: 30 cm
- Width: 20 cm
- Height: 20 cm
The smallest dimension here is 20 cm. Therefore, the diameter of the largest sphere that can fit in the box is 20 cm.
The radius \( r \) of the sphere is half of the diameter:
\[ r = \frac{d}{2} = \frac{20 , \text{cm}}{2} = 10 , \text{cm} \]
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
\[ \boxed{400\pi} , \text{cm}^2 \]