https://file.aiquickdraw.com/mathbot/file/1738096729873nk9q3tk0.jpg

To find the surface area of the given 3D shape, we need to determine the areas of all its faces and sum them up.

### Given dimensions:
- **Top width** = 10 mm  
- **Height** = 5.6 mm  
- **Bottom widths** = 8 mm and 6 mm  

The shape appears to be an L-shaped cross-section or an angular channel. Let's calculate the surface area by identifying each face.

#### **Step 1: Identify the Faces**
The object has the following surfaces:
1. **Two triangular faces (front and back)**
2. **Three rectangular faces (bottom, inclined, and back vertical face)**

#### **Step 2: Calculate Individual Areas**
- **Triangular Faces (Front and Back, 2 identical faces)**
  - Base = 10 mm
  - Height = 5.6 mm
  - Area of one triangle = \( \frac{1}{2} \times 10 \times 5.6 = 28 \) mm²
  - Total for two triangles = \( 2 \times 28 = 56 \) mm²

- **Bottom Rectangular Face**
  - Width = 8 mm
  - Length = 6 mm
  - Area = \( 8 \times 6 = 48 \) mm²

- **Inclined Rectangular Face**
  - Hypotenuse length = calculated using Pythagoras’ theorem:
    \[
    \sqrt{(10 - 8)^2 + 5.6^2} = \sqrt{2^2 + 5.6^2} = \sqrt{4 + 31.36} = \sqrt{35.36} \approx 5.95 \text{ mm}
    \]
  - Width = 6 mm
  - Area = \( 5.95 \times 6 = 35.7 \) mm²

- **Back Vertical Rectangular Face**
  - Width = 8 mm
  - Height = 5.6 mm
  - Area = \( 8 \times 5.6 = 44.8 \) mm²

#### **Step 3: Total Surface Area**
\[
\text{Total} = 56 + 48 + 35.7 + 44.8 = 184.5 \text{ mm}^2
\]

### **Final Answer:**
\[
\mathbf{184.5} \text{ mm}^2
\]

Would you like a breakdown or any clarifications?  

Here are 5 related questions to explore further:  
1. How would the surface area change if the bottom width increased?  
2. How do you calculate the volume of this shape?  
3. How does the Pythagorean theorem help in finding surface areas?  
4. What is the lateral surface area of this shape?  
5. Can we approximate this shape using simpler 3D objects like a prism?

**Tip:** When solving surface area problems, always list and visualize each face separately to avoid missing any.

Find the surface area of the given 3D shape.

Learn how to calculate the surface area of a 3D angular shape using geometry principles. This step-by-step guide covers triangular and rectangular face calculations, including the Pythagorean theorem application. Suitable for students in Grades 8-10.

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