https://file.aiquickdraw.com/mathbot/file/172561760572280n4ui0n.jpg

The image describes a compound prism made from two rectangles, and the task is to calculate the surface area of the prism. Here’s the step-by-step approach to solving the surface area:

### Step 1: Understand the dimensions
The shape is a combination of two rectangular prisms. Let's extract the dimensions:

- The larger rectangle's dimensions are:
  - Height: 15 cm
  - Width: 12 cm
  - Depth: 6 cm

- The smaller rectangle on top has dimensions:
  - Height: 4 cm
  - Width: 6 cm (as this matches the depth of the lower rectangle)
  - Depth: 6 cm

### Step 2: Break down the surface area
The total surface area is the sum of the surface areas of each prism, but we need to subtract any faces where the two rectangles meet, as these areas are not exposed.

#### Surface area of the larger prism
- **Front and back faces**: \( 2 \times (12 \times 15) = 2 \times 180 = 360 \, \text{cm}^2 \)
- **Top face of the larger prism**: This is partly covered by the smaller rectangle, so we only consider the uncovered part.
  - Uncovered part: \( 12 \, \text{cm} \times (15 - 4) = 12 \times 11 = 132 \, \text{cm}^2 \)
- **Bottom face**: \( 12 \, \text{cm} \times 6 \, \text{cm} = 72 \, \text{cm}^2 \)
- **Left and right faces**:
  - Left face: \( 6 \, \text{cm} \times 15 \, \text{cm} = 90 \, \text{cm}^2 \)
  - Right face: Same as left, so \( 90 \, \text{cm}^2 \)

#### Surface area of the smaller prism
- **Front and back faces**: \( 2 \times (4 \times 6) = 48 \, \text{cm}^2 \)
- **Top face**: \( 6 \times 6 = 36 \, \text{cm}^2 \)
- **Left and right faces**:
  - Left face: \( 4 \, \text{cm} \times 6 \, \text{cm} = 24 \, \text{cm}^2 \)
  - Right face: \( 4 \times 6 = 24 \, \text{cm}^2 \)

### Step 3: Total surface area
Now, sum up the areas of each face, subtracting the parts where the two prisms meet:

1. Larger prism surface area (excluding covered part):
   - Front and back: \( 360 \, \text{cm}^2 \)
   - Top (uncovered): \( 132 \, \text{cm}^2 \)
   - Bottom: \( 72 \, \text{cm}^2 \)
   - Left and right: \( 90 + 90 = 180 \, \text{cm}^2 \)
   
2. Smaller prism surface area:
   - Front and back: \( 48 \, \text{cm}^2 \)
   - Top: \( 36 \, \text{cm}^2 \)
   - Left and right: \( 24 + 24 = 48 \, \text{cm}^2 \)

### Step 4: Final Calculation
Let’s sum these areas.

I'll calculate the final surface area now.The total surface area of the compound prism is **876 cm²**.

Do you need further clarification on any part of this calculation?

Here are some related questions you might find helpful:
1. How would the surface area change if the height of the smaller prism increased?
2. What is the volume of this compound prism?
3. How would the surface area calculation change if there were a third rectangular prism on top?
4. Can the surface area be calculated using a different method or formula?
5. How is the surface area of a cylinder compared to this type of prism?

**Tip:** When calculating the surface area of composite shapes, always make sure to account for the overlapping or hidden faces to avoid double-counting those areas.

The cross-section of the prism below is a compound shape formed of two rectangles. Calculate the surface area of this prism. Give your answer in cm².

Learn how to calculate the surface area of a compound prism formed by two rectangular shapes with detailed steps. This example shows how to break down the shape, extract dimensions, and apply surface area formulas. Suitable for students in Grades 8-10.

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