what is the surface of the right rectangle prism? show your work

To find the surface area of a right rectangular prism, you need to calculate the area of all six faces of the prism and then sum them up. A right rectangular prism has three pairs of identical faces.

### Let's break down the process:

1. **Identify the dimensions of the prism:**
   - Let \( l \) be the length.
   - Let \( w \) be the width.
   - Let \( h \) be the height.

2. **Formula for the surface area:**
   The surface area \( A \) of a right rectangular prism is given by the formula:
   \[
   A = 2lw + 2lh + 2wh
   \]
   This accounts for the areas of the three pairs of identical faces.

3. **Detailed Calculation:**
   - The area of the two faces with dimensions \( l \times w \) is \( 2lw \).
   - The area of the two faces with dimensions \( l \times h \) is \( 2lh \).
   - The area of the two faces with dimensions \( w \times h \) is \( 2wh \).

   Therefore, the total surface area \( A \) is:
   \[
   A = 2lw + 2lh + 2wh
   \]

4. **Example Calculation:**
   Suppose the dimensions of the prism are:
   - Length (\( l \)) = 5 units
   - Width (\( w \)) = 3 units
   - Height (\( h \)) = 4 units

   Plug these values into the formula:
   \[
   A = 2(5 \times 3) + 2(5 \times 4) + 2(3 \times 4)
   \]
   \[
   A = 2(15) + 2(20) + 2(12)
   \]
   \[
   A = 30 + 40 + 24 = 94 \text{ square units}
   \]

   So, the surface area of this right rectangular prism is 94 square units.

Would you like any further details or have any questions? 

Here are 5 related questions you might find interesting:
1. How does the surface area change if one of the dimensions doubles?
2. What is the surface area if the dimensions are all equal (i.e., a cube)?
3. How do you find the volume of a right rectangular prism?
4. What happens to the surface area if only the height is tripled?
5. Can you derive the formula for the surface area of a prism from the general shape of the faces?

**Tip:** Always double-check the units of your measurements to ensure consistency when calculating surface areas.

Discover how to calculate the surface area of a right rectangular prism with this detailed guide. Follow along as we break down the process using the formula A = 2lw + 2lh + 2wh, suitable for students in Grades 5-7.

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