< b="" /> Problem Statement: < />
A cube is inscribed in a sphere with a radius of 8 cm. We need to find the total surface area of the cube.

< b="" /> Solution: < />
To find the total surface area of the cube, we need to determine the length of one side of the cube. Let's assume the length of one side of the cube is 'a' cm.

< b="" /> Identifying Key Points: < />

- The cube is inscribed in a sphere, which means the sphere's diameter is equal to the length of the diagonal of the cube.
- The diameter of the sphere is given as 8 cm.
- The length of the diagonal of the cube is equal to the length of the side multiplied by the square root of 3.

< b="" /> Finding the Length of the Diagonal: < />
The diameter of the sphere is given as 8 cm, which means the radius is half of the diameter, i.e., 4 cm. The length of the diagonal of the cube is equal to the diameter of the sphere.

Therefore, the length of the diagonal of the cube = 8 cm.

Since the length of the diagonal of the cube is equal to the length of the side multiplied by the square root of 3, we can write the equation as:

a * √3 = 8

< b="" /> Finding the Length of One Side of the Cube: < />
Dividing both sides of the equation by √3, we get:

a = 8 / √3

< b="" /> Calculating the Length of One Side of the Cube: < />
Using a calculator, we can calculate the value of a as approximately 4.6188 cm.

< b="" /> Finding the Total Surface Area of the Cube: < />
The total surface area of a cube is given by the formula: 6 * a^2.

Plugging in the value of 'a', we get:

Total surface area = 6 * (4.6188)^2

Calculating the above expression, we get the total surface area of the cube as approximately 512 sq. cm.

Therefore, the correct answer is '512'.