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The problem involves a cube with edge length 4 cm, which is cut by a plane consisting of two diagonally opposite edges of the cube. The task is to find the area of the section formed by this plane. The answer provided is \( A = 16 \sqrt{2} \, \text{cm}^2 \).

### Solution:

The cube's edge is 4 cm, and when a plane passes through two opposite diagonals, the resulting section forms a **rhombus**. Here's how to approach it:

1. **Diagonals of the Rhombus:**
   - The rhombus is formed by the intersection of the plane with the cube.
   - The diagonals of the rhombus correspond to the face diagonals of the cube.
   - A face diagonal of the cube can be calculated using the Pythagorean theorem:
     \[
     \text{Face Diagonal} = \sqrt{4^2 + 4^2} = 4\sqrt{2} \, \text{cm}.
     \]

2. **Area of a Rhombus:**
   The area \( A \) of a rhombus can be found using the formula:
   \[
   A = \frac{1}{2} \times d_1 \times d_2
   \]
   where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

   - In this case, the diagonals are both \( 4\sqrt{2} \, \text{cm} \) because the rhombus is symmetric, formed by the two diagonally opposite edges of the cube.

3. **Calculation:**
   \[
   A = \frac{1}{2} \times 4\sqrt{2} \times 4\sqrt{2}
   \]
   \[
   A = \frac{1}{2} \times 16 \times 2 = 16\sqrt{2} \, \text{cm}^2.
   \]

Thus, the area of the section formed is \( 16\sqrt{2} \, \text{cm}^2 \), which matches the answer provided.

Would you like further details or have any questions?

### Related Questions:
1. How can we find the volume of a cube with a given edge length?
2. What is the formula for the surface area of a cube?
3. How do you calculate the diagonal of a cube (not a face diagonal)?
4. What is the geometric shape of the section formed when a cube is cut by different types of planes?
5. How does the cutting plane affect the section area if the angle between the plane and cube edges changes?

**Tip:** For any 3D object like a cube, diagonals (both face and space diagonals) are important in many cutting and sectioning problems.

A cube of edge 4 cm is cut by a plane consisting of two diagonally opposite edges of the cube. Find the area of the section formed. (Answer: A = 16√2 cm²).

Learn how to find the area of a rhombus-shaped section formed by cutting a cube with a plane through two diagonally opposite edges. This problem covers concepts like the Pythagorean theorem and the formula for the area of a rhombus. Suitable for Grades 10-12.

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