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To find the volume of the frustum in the given problem, we can follow this approach:

### Formula for the volume of a frustum of a pyramid:
The volume \( V \) of a frustum of a pyramid is given by:
\[
V = \frac{1}{3} h \left( A_1 + A_2 + \sqrt{A_1 \cdot A_2} \right)
\]
where:
- \( h \) is the height of the frustum,
- \( A_1 \) is the area of the base of the larger section,
- \( A_2 \) is the area of the base of the smaller section.

### Step-by-step calculation:

1. **Dimensions from the diagram**:
   - The base of the larger pyramid section is a rectangle with sides \( 7 \, \text{cm} \) and \( 14 \, \text{cm} \), so its area \( A_1 \) is:
     \[
     A_1 = 7 \times 14 = 98 \, \text{cm}^2
     \]
   - The base of the smaller pyramid section (top) is a rectangle with sides \( 3 \, \text{cm} \) and \( 6 \, \text{cm} \), so its area \( A_2 \) is:
     \[
     A_2 = 3 \times 6 = 18 \, \text{cm}^2
     \]
   - The height of the frustum is given as \( 12 \, \text{cm} \).

2. **Plug these values into the formula**:
   \[
   V = \frac{1}{3} \times 12 \left( 98 + 18 + \sqrt{98 \times 18} \right)
   \]
   First, calculate \( \sqrt{98 \times 18} \):
   \[
   \sqrt{98 \times 18} = \sqrt{1764} = 42
   \]
   Now, substitute this back into the volume formula:
   \[
   V = \frac{1}{3} \times 12 \left( 98 + 18 + 42 \right)
   \]
   Simplify the expression inside the parentheses:
   \[
   V = \frac{1}{3} \times 12 \times 158
   \]
   \[
   V = 4 \times 158 = 632 \, \text{cm}^3
   \]

### Final Answer:
The volume of the frustum is \( 632 \, \text{cm}^3 \).

Would you like more details on any step or have any further questions?

Here are five related questions you might want to explore:
1. How would the formula change if the frustum had a triangular base?
2. What would the volume be if the smaller pyramid was removed entirely?
3. How does the volume of the original pyramid compare to that of the frustum?
4. How can the formula be adapted for a frustum with non-parallel sides?
5. What real-world applications can this volume calculation method be used for?

**Tip:** Always double-check units to ensure accuracy when performing calculations like volume!

Work out the volume of the frustum of a rectangular-based pyramid with the following dimensions: larger base (7 cm by 14 cm), smaller base (3 cm by 6 cm), and height (12 cm).

Learn how to calculate the volume of a frustum of a rectangular-based pyramid using geometry. This example involves a large base of 7 cm by 14 cm, a smaller base of 3 cm by 6 cm, and a height of 12 cm. Follow step-by-step instructions to find the volume using the frustum volume formula.

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