We are tasked with determining the sum of all possible surface areas of a cube where two of its vertices are given: $(1, 2, 0)$ and $(3, 3, 2)$. To solve this, we need to find the edge length of the cube, and from that, we can calculate the surface area.

Step 1: Calculate the distance between the two vertices

The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in 3-dimensional space is given by the formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Substituting the coordinates of the two vertices $(1, 2, 0)$ and $(3, 3, 2)$:

$d = \sqrt{(3 - 1)^2 + (3 - 2)^2 + (2 - 0)^2}$ $d = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$

So, the distance between the two vertices is 3.

Step 2: Determine the edge length of the cube

In a cube, the distance between any two opposite vertices (also known as the space diagonal) is related to the edge length $a$ by the following formula:

$d_{\text{diagonal}} = a\sqrt{3}$

We know that the distance between two vertices is 3, so:

$a\sqrt{3} = 3$

Solving for $a$:

$a = \frac{3}{\sqrt{3}} = \sqrt{3}$

Thus, the edge length of the cube is $\sqrt{3}$.

Step 3: Calculate the surface area of the cube

The surface area $A$ of a cube with edge length $a$ is given by the formula:

$A = 6a^2$

Substituting $a = \sqrt{3}$:

$A = 6(\sqrt{3})^2 = 6 \times 3 = 18$

Final Answer:

The surface area of the cube is $18$ square units.

Do you have any questions or need further details?

Here are some related questions you might find useful:

1. How do you find the volume of a cube if the edge length is known?
2. What is the relationship between the diagonal of a cube and its edge length?
3. Can the surface area formula be applied to other polyhedra?
4. How does changing the edge length affect the surface area of a cube?
5. What are the properties of the space diagonal in other 3D geometric shapes?

Tip: The space diagonal of a cube is longer than the face diagonal by a factor of $\sqrt{3}$. This can help distinguish between different types of diagonals in geometry problems.