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