To determine the surface area of the largest sphere that can fit inside a cube with dimensions of 20 cm × 20 cm × 20 cm, we first identify the largest diameter of the sphere. The largest sphere that can fit inside the cube will have a diameter equal to the side length of the cube.
Since the side length of the cube is 20 cm, the diameter of the sphere is also 20 cm. Thus, the radius of the sphere \( r \) is half of the diameter:
\[ r = \frac{20 , \text{cm}}{2} = 10 , \text{cm} \]
The surface area \( A \) of a sphere is calculated using the formula:
\[ A = 4\pi r^2 \]
Substituting the radius:
\[ A = 4\pi (10 , \text{cm})^2 \] \[ A = 4\pi (100 , \text{cm}^2) \] \[ A = 400\pi , \text{cm}^2 \]
To give a numerical approximation, using \( \pi \approx 3.14 \):
\[ A \approx 400 \times 3.14 , \text{cm}^2 \approx 1256 , \text{cm}^2 \]
Thus, the surface area of the largest sphere that can fit in the cube is:
\[ \boxed{400\pi , \text{cm}^2} \quad \text{or approximately } 1256 , \text{cm}^2. \]