https://file.aiquickdraw.com/mathbot/file/1723968471986lgu3escr.jpg

To find the surface area of the prism depicted in the image, we need to calculate the area of each face and sum them up.

### Dimensions:
- **Height**: 4 cm
- **Depth**: 8 cm
- **Base width**: 16 cm

### Steps to calculate the surface area:
1. **Identify the faces:**
   - **Top and bottom faces:** These are the rectangles formed by the base (16 cm) and depth (8 cm).
   - **Front face:** This is a composite face consisting of two rectangles.
   - **Back face:** Similar to the front face.
   - **Side faces (left and right):** Each consists of two smaller rectangles.

2. **Calculate the area of each face:**
   - **Top and Bottom faces** (2 rectangles):
     \[
     \text{Area of each} = \text{Base} \times \text{Depth} = 16 \text{ cm} \times 8 \text{ cm} = 128 \text{ cm}^2
     \]
   - **Front and Back faces** (each has two parts):
     - **Upper rectangle**: Height \(= 4 \text{ cm}\), Width \(= 8 \text{ cm}\)
     \[
     \text{Area} = 4 \text{ cm} \times 8 \text{ cm} = 32 \text{ cm}^2
     \]
     - **Lower rectangle**: Height \(= 8 \text{ cm}\), Width \(= 16 \text{ cm}\)
     \[
     \text{Area} = 8 \text{ cm} \times 16 \text{ cm} = 128 \text{ cm}^2
     \]
     \[
     \text{Total area of Front face} = 32 \text{ cm}^2 + 128 \text{ cm}^2 = 160 \text{ cm}^2
     \]
     Since the front and back faces are identical:
     \[
     \text{Total area of Front and Back faces} = 2 \times 160 \text{ cm}^2 = 320 \text{ cm}^2
     \]
   - **Left and Right faces** (each has two parts):
     - **Left small rectangle**: Height \(= 4 \text{ cm}\), Depth \(= 4 \text{ cm}\)
     \[
     \text{Area} = 4 \text{ cm} \times 4 \text{ cm} = 16 \text{ cm}^2
     \]
     - **Left large rectangle**: Height \(= 8 \text{ cm}\), Depth \(= 4 \text{ cm}\)
     \[
     \text{Area} = 8 \text{ cm} \times 4 \text{ cm} = 32 \text{ cm}^2
     \]
     \[
     \text{Total area of Left face} = 16 \text{ cm}^2 + 32 \text{ cm}^2 = 48 \text{ cm}^2
     \]
     Since the left and right faces are identical:
     \[
     \text{Total area of Left and Right faces} = 2 \times 48 \text{ cm}^2 = 96 \text{ cm}^2
     \]

3. **Sum all the areas to get the total surface area:**
\[
\text{Total Surface Area} = 128 \text{ cm}^2 + 128 \text{ cm}^2 + 320 \text{ cm}^2 + 96 \text{ cm}^2 = 672 \text{ cm}^2
\]

### The surface area of the prism is \(672 \, \text{cm}^2\).

Would you like further details or have any questions?

Here are five related questions you might consider:
1. What would be the surface area if the prism's depth were doubled?
2. How would the surface area change if the height were increased by 2 cm?
3. What is the volume of this prism?
4. How does the surface area relate to the volume in such prisms?
5. What if the base was triangular instead of rectangular; how would you calculate the surface area?

**Tip:** When calculating the surface area of complex shapes, break down the shape into simpler parts and find the area of each part separately.

Learn how to calculate the surface area of a prism with dimensions 16 cm x 8 cm x 4 cm. This problem involves finding the areas of rectangular faces and applying the formula for surface area of a prism. Suitable for students in Grades 6-8.

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