Determine the sum of all possible surface area of a cube  two of whose vertices are ( 1, 2, 0 ) and (3, 3, 2 )

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.

Determine the sum of all possible surface areas of a cube two of whose vertices are (1, 2, 0) and (3, 3, 2).

Learn how to calculate the surface area of a cube when two vertices are given. This problem uses the distance formula and cube diagonal-edge length relationships to find the edge length and surface area. Suitable for high school students in Grades 10-12.

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