Question

Calculate surface area of two connected rectangular prisms

Answer 1

To calculate the surface area of two connected rectangular prisms, we need to consider the dimensions of each prism and how they connect (i.e., whether they share a face, edge, or vertex).

### Steps to Calculate Surface Area:

1. **Determine the Dimensions**: Let’s denote the dimensions of the first prism as \( l_1 \), \( w_1 \), and \( h_1 \), and the dimensions of the second prism as \( l_2 \), \( w_2 \), and \( h_2 \).

2. **Calculate Surface Area for Each Prism**:
- The surface area \( S \) of a rectangular prism is given by the formula:
\[
S = 2(l \cdot w + l \cdot h + w \cdot h)
\]
- For the first prism:
\[
S_1 = 2(l_1 \cdot w_1 + l_1 \cdot h_1 + w_1 \cdot h_1)
\]
- For the second prism:
\[
S_2 = 2(l_2 \cdot w_2 + l_2 \cdot h_2 + w_2 \cdot h_2)
\]

3. **Account for the Shared Area**: When the two prisms are connected, they will share some surface area.

- If they are connected along a face, calculate the area of the shared face. For example, if they are connected along a face of dimensions \( a \) by \( b \), the shared area \( A_{\text{shared}} \) is:
\[
A_{\text{shared}} = a \cdot b
\]
- Subtract the shared area from the total surface area of the two prisms:
\[
A_{\text{total}} = S_1 + S_2 - A_{\text{shared}}
\]

### Example Calculation:
Let’s consider two rectangular prisms:
- **Prism 1**: \( l_1 = 3 \), \( w_1 = 4 \), \( h_1 = 5 \)
- **Prism 2**: \( l_2 = 2 \), \( w_2 = 3 \), \( h_2 = 6 \)
- Assume they connect along a face of size \( 2 \times 3 \).

#### Surface Area Calculations:
1. **For Prism 1**:
\[
S_1 = 2(3 \cdot 4 + 3 \cdot 5 + 4 \cdot 5) = 2(12 + 15 + 20) = 2(47) = 94
\]

2. **For Prism 2**:
\[
S_2 = 2(2 \cdot 3 + 2 \cdot 6 + 3 \cdot 6) = 2(6 + 12 + 18) = 2(36) = 72
\]

3. **Shared Area**:
\[
A_{\text{shared}} = 2 \cdot 3 = 6
\]

4. **Total Surface Area**:
\[
A_{\text{total}} = S_1 + S_2 - A_{\text{shared}} = 94 + 72 - 6 = 160
\]

Thus, the total surface area of the two connected rectangular prisms is **160 square units**.