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What is the surface area of the three-dimensional figure?

To determine the surface area of the given three-dimensional figure, we need to analyze the shape and dimensions of the individual blocks. The figure appears to be made of three rectangular prisms combined together. Let's break it down step by step.

### Step 1: Identify and label the blocks

Let's label the blocks as follows:
- **Block A**: The topmost block.
- **Block B**: The middle block (left).
- **Block C**: The bottom block (right).

### Step 2: Determine the dimensions of each block
- **Block A**:
  - Length: 5 inches
  - Width: 4 inches
  - Height: 6 inches

- **Block B**:
  - Length: 5 inches
  - Width: 5 inches
  - Height: 6 inches

- **Block C**:
  - Length: 5 inches
  - Width: 5 inches
  - Height: 6 inches

### Step 3: Calculate the surface area of each block

#### Surface area of Block A:
\[ 2 \times (L \times W + L \times H + W \times H) \]
\[ = 2 \times (5 \times 4 + 5 \times 6 + 4 \times 6) \]
\[ = 2 \times (20 + 30 + 24) \]
\[ = 2 \times 74 \]
\[ = 148 \, \text{square inches} \]

#### Surface area of Block B:
\[ 2 \times (L \times W + L \times H + W \times H) \]
\[ = 2 \times (5 \times 5 + 5 \times 6 + 5 \times 6) \]
\[ = 2 \times (25 + 30 + 30) \]
\[ = 2 \times 85 \]
\[ = 170 \, \text{square inches} \]

#### Surface area of Block C:
\[ 2 \times (L \times W + L \times H + W \times H) \]
\[ = 2 \times (5 \times 5 + 5 \times 6 + 5 \times 6) \]
\[ = 2 \times (25 + 30 + 30) \]
\[ = 2 \times 85 \]
\[ = 170 \, \text{square inches} \]

### Step 4: Adjust for overlapping surfaces

We must subtract the areas of the faces that are hidden because they are between blocks:
- The face between Block A and Block B:
  \[ 5 \, \text{in} \times 6 \, \text{in} = 30 \, \text{square inches} \]
  
- The face between Block B and Block C:
  \[ 5 \, \text{in} \times 5 \, \text{in} = 25 \, \text{square inches} \]
  
- The face between Block A and Block C (top block overlapping the other two):
  \[ 5 \, \text{in} \times 4 \, \text{in} = 20 \, \text{square inches} \]

### Step 5: Calculate the total surface area
\[ \text{Total Surface Area} = (\text{Surface Area of A} + \text{Surface Area of B} + \text{Surface Area of C}) - (\text{Overlap between A and B} + \text{Overlap between B and C} + \text{Overlap between A and C}) \]
\[ = (148 + 170 + 170) - (30 + 25 + 20) \]
\[ = 488 - 75 \]
\[ = 413 \, \text{square inches} \]

Thus, the surface area of the three-dimensional figure is **413 square inches**.

Would you like more details on any specific part of the calculation, or do you have any other questions?

Here are some related questions to expand your understanding:
1. How would you find the volume of the three-dimensional figure?
2. How does the surface area change if the height of Block A is increased?
3. What if the length of Block B is reduced by 1 inch? How does that affect the surface area?
4. How can you calculate the surface area of an irregular solid composed of different shapes?
5. What are some real-life applications of calculating surface areas of three-dimensional figures?
6. Can you explain how the concept of surface area is applied in material science?
7. How would the surface area change if a new block is added on top of Block A?
8. How do you calculate the surface area of a composite shape that includes curved surfaces?

### Tip
When calculating surface areas, always remember to consider and subtract any overlapping or hidden faces between the shapes to avoid counting them multiple times.

Learn how to calculate the surface area of a three-dimensional figure composed of three rectangular prisms. Step-by-step guide with detailed dimensions and formula application. Suitable for students in Grades 7-9.

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