To determine the surface area of the largest sphere that can fit inside a cube, we first need to find the radius 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 cube has side lengths of 20 cm, the diameter of the largest sphere is also 20 cm.
From this, we can find the radius of the sphere:
\[ \text{Radius (r)} = \frac{\text{Diameter}}{2} = \frac{20 , \text{cm}}{2} = 10 , \text{cm} \]
Next, we calculate the surface area of the sphere using the formula for the surface area of a sphere, which is given by:
\[ \text{Surface Area} = 4\pi r^2 \]
Substituting the radius we found:
\[ \text{Surface Area} = 4\pi (10 , \text{cm})^2 \]
Calculating \( (10 , \text{cm})^2 \):
\[ (10 , \text{cm})^2 = 100 , \text{cm}^2 \]
Now substituting back into the surface area formula:
\[ \text{Surface Area} = 4\pi \times 100 , \text{cm}^2 = 400\pi , \text{cm}^2 \]
Thus, the surface area of the largest sphere that can fit inside the cube is:
\[ \boxed{400\pi , \text{cm}^2} \]
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