A cube having each side of unit length is cut into two parts by plane through two diagonals of two opposite faces. What is the total surface area of each of these parts ?

Options

A. `3+sqrt(2)` sq. units
B. `2+sqrt(3)` sq. units
C. `3sqrt(2)` sq. units
D. 3 sq. units

Answer

To find the total surface area of each part of the cube after it has been cut by a plane through two diagonals of two opposite faces, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Cube's Dimensions**:
- The cube has a side length of 1 unit. Therefore, the dimensions of the cube are 1 unit x 1 unit x 1 unit.

2. **Calculate the Total Surface Area of the Cube**:
- The formula for the total surface area (SA) of a cube is given by:
\[ SA = 6 \times (\text{side length})^2 \]
- Substituting the side length:
\[ SA = 6 \times (1)^2 = 6 \text{ square units} \]

3. **Identify the Plane Cutting the Cube**:
- The cube is cut by a plane through the diagonals of two opposite faces. This means we need to understand how this cut affects the surface area.

4. **Determine the New Surface Areas Created by the Cut**:
- The cut creates two new triangular faces. Each triangular face is formed by the diagonals of the square faces of the cube.
- The area of one triangular face can be calculated using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- The base and height of the triangle formed by the diagonal of a square face (1 unit x 1 unit) are both equal to 1 unit. Therefore, the area of one triangular face is:
\[ \text{Area} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \text{ square units} \]
- Since there are two triangular faces created by the cut, the total area contributed by these triangular faces is:
\[ 2 \times \frac{1}{2} = 1 \text{ square unit} \]

5. **Calculate the Total Surface Area of Each Part**:
- Each part of the cube will have half of the original surface area plus the area of the new triangular faces. Therefore, the total surface area of one part is:
\[ \text{Total Surface Area of One Part} = \frac{6}{2} + 1 = 3 + 1 = 4 \text{ square units} \]

### Final Answer:
The total surface area of each part after the cube is cut is **4 square units**.