https://file.aiquickdraw.com/mathbot/file/1725165286146fg65rkcn.jpg

To calculate the volume of the L-shaped 3D object shown in the image, we can split the shape into two rectangular prisms and find their volumes separately. Then, we will sum the volumes of these two prisms to get the total volume.

### Step-by-Step Solution:

1. **Identify and Break Down the Shape:**

   The L-shaped object can be broken down into two rectangular prisms:
   
   - **Prism 1 (Vertical Part):** 
     - Height = 10 mm
     - Width = 3 mm
     - Depth = 3 mm

   - **Prism 2 (Horizontal Part):** 
     - Height = 4 mm
     - Width = 6 mm
     - Depth = 3 mm

2. **Calculate the Volume of Each Prism:**

   - **Volume of Prism 1:**
   \[
   V_1 = \text{Height} \times \text{Width} \times \text{Depth} = 10 \, \text{mm} \times 3 \, \text{mm} \times 3 \, \text{mm} = 90 \, \text{mm}^3
   \]

   - **Volume of Prism 2:**
   \[
   V_2 = \text{Height} \times \text{Width} \times \text{Depth} = 4 \, \text{mm} \times 6 \, \text{mm} \times 3 \, \text{mm} = 72 \, \text{mm}^3
   \]

3. **Calculate the Total Volume:**

   Since the two prisms overlap at a part of the object, we need to subtract the overlapping volume to get the correct total volume.

   - **Overlapping Volume:**
     - Height = 4 mm (the overlapping height with Prism 2)
     - Width = 3 mm
     - Depth = 3 mm
   \[
   V_{\text{overlap}} = 4 \, \text{mm} \times 3 \, \text{mm} \times 3 \, \text{mm} = 36 \, \text{mm}^3
   \]

4. **Find the Correct Total Volume:**

   \[
   V_{\text{total}} = V_1 + V_2 - V_{\text{overlap}} = 90 \, \text{mm}^3 + 72 \, \text{mm}^3 - 36 \, \text{mm}^3 = 126 \, \text{mm}^3
   \]

### Final Answer:

The total volume of the L-shaped object is **126 mm³**.

Would you like more details or have any questions about this calculation?

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Here are 5 related questions that might interest you:

1. How do you find the surface area of this L-shaped object?
2. What is the volume of a similar object if all dimensions are doubled?
3. How would the calculation change if the shape were hollow instead of solid?
4. What is the center of mass of this L-shaped object assuming uniform density?
5. How does the overlapping volume affect the total volume in more complex shapes?

**Tip:** When calculating volumes of composite shapes, always check for any overlapping regions to avoid over-counting their volumes.

Learn how to calculate the volume of an L-shaped 3D object using a detailed step-by-step solution. This problem involves volume calculations of rectangular prisms and handling overlapping volumes. Suitable for advanced high school students and anyone interested in solid geometry.

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