A cube whose edge is 20 cm long, has circles on each of its faces painted black. What is the total area of the unpainted surface of the cube, if the circles are of the largest possible areas?

Options

A. 90.72 `cm^(2)`
B. 256.72 `cm^(2)`
C. 330.3 `cm^(2)`
D. 516 `cm^(2)`

Answer

To find the total area of the unpainted surface of a cube with circles painted on each face, we can follow these steps:

### Step 1: Calculate the surface area of the cube.
The formula for the total surface area (TSA) of a cube is given by:
\[ \text{TSA} = 6a^2 \]
where \(a\) is the length of an edge of the cube.

Given that the edge of the cube is 20 cm:
\[ \text{TSA} = 6 \times (20 \, \text{cm})^2 = 6 \times 400 \, \text{cm}^2 = 2400 \, \text{cm}^2 \]

### Step 2: Determine the radius of the circles on each face.
Since the circles are of the largest possible area, they will be inscribed in the square face of the cube. The diameter of each circle will be equal to the side length of the cube.

Thus, the diameter of the circle is 20 cm, which gives us a radius \(r\) of:
\[ r = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

### Step 3: Calculate the area of one circle.
The area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi \times (10 \, \text{cm})^2 = \pi \times 100 \, \text{cm}^2 \]
Using \(\pi \approx \frac{22}{7}\) or \(3.14\):
\[ A \approx 3.14 \times 100 \, \text{cm}^2 = 314 \, \text{cm}^2 \]

### Step 4: Calculate the total area of the circles on all faces.
Since there are 6 faces on the cube, the total area of the circles is:
\[ \text{Total area of circles} = 6 \times A = 6 \times 314 \, \text{cm}^2 = 1884 \, \text{cm}^2 \]

### Step 5: Calculate the unpainted surface area of the cube.
To find the unpainted surface area, we subtract the total area of the circles from the total surface area of the cube:
\[ \text{Unpainted area} = \text{TSA} - \text{Total area of circles} \]
\[ \text{Unpainted area} = 2400 \, \text{cm}^2 - 1884 \, \text{cm}^2 = 516 \, \text{cm}^2 \]

### Final Answer:
The total area of the unpainted surface of the cube is:
\[ \boxed{516 \, \text{cm}^2} \]