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To calculate the surface area of the frustum, we need to consider:

1. **The top and bottom rectangular faces of the frustum**, whose dimensions are provided.
2. **The slanted sides**, which form trapezoids. There are four slanted sides in total, each corresponding to one side of the rectangular base.

### Information from the image:
- **Top rectangular face dimensions**: \( 80 \, \text{mm} \times 16 \, \text{mm} \)
- **Bottom rectangular face dimensions**: \( 200 \, \text{mm} \times 80 \, \text{mm} \)
- **Height of the frustum**: \( 15 \, \text{mm} \)

We need to calculate the surface areas of each component:

#### 1. Surface area of the **top and bottom faces**:
- **Top face area**: \( A_{\text{top}} = 80 \times 16 \)
- **Bottom face area**: \( A_{\text{bottom}} = 200 \times 80 \)

#### 2. Surface area of the **slanted trapezoidal faces**:
The area of each trapezoid depends on the height of the frustum and the lengths of the corresponding sides of the top and bottom faces. The slant height can be determined using the Pythagorean theorem since the slanted sides are the hypotenuses of right triangles formed by half the difference in side lengths of the top and bottom and the height of the frustum.

I will calculate these values step-by-step.The total surface area of the frustum is approximately **37,989.8 mm²** when rounded to one decimal place.

Would you like a more detailed explanation of the steps? Here are 5 related questions to expand on this:

1. How is the slant height of a frustum calculated?
2. What is the formula for the surface area of a trapezoidal face?
3. How would the surface area change if the frustum height were different?
4. Can we apply this approach to other types of pyramids (e.g., square-based pyramids)?
5. What is the volume of this frustum, and how can it be calculated?

**Tip**: When working with frustums, always remember that the slant height plays a key role in determining the area of the trapezoidal sides.

What is the surface area of the frustum formed by a rectangular-based pyramid, with a height of 15 mm and top and bottom rectangular faces measuring 80 mm x 16 mm and 200 mm x 80 mm respectively?

Calculate the surface area of a frustum derived from a rectangular-based pyramid, with given dimensions of the top and bottom faces. This solution involves geometry concepts such as trapezoidal areas and the Pythagorean theorem, suitable for students in Grades 10-12.

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