To calculate the surface area of the frustum, we need to address two components: the lateral (side) surface area and the areas of the top and bottom square faces.

1. Base Areas:

• The bottom face of the frustum is a square with a side length of 48 mm, so its area is: $A_{\text{bottom}} = 48^2 = 2304 \text{ mm}^2$

• The top face is a square with a side length of 37 mm, so its area is: $A_{\text{top}} = 37^2 = 1369 \text{ mm}^2$

2. Lateral Surface Area (Trapezoidal Faces):

Each lateral face is a trapezium (trapezoid), and we will compute the area of one trapezoidal face and multiply by 4 (since there are four lateral faces).

For one trapezium, the parallel sides are the heights of the top and bottom squares:

• Larger base: $48$ mm
• Smaller base: $37$ mm
  The slant height of each trapezoid face is given as 37 mm.

The area of a trapezium is: $A_{\text{trapezium}} = \frac{1}{2} \times (b_1 + b_2) \times h$ Where:

• $b_1 = 48$ mm (larger base),
• $b_2 = 37$ mm (smaller base),
• $h = 37$ mm (slant height).

Now, calculate the area: $A_{\text{trapezium}} = \frac{1}{2} \times (48 + 37) \times 37 = \frac{1}{2} \times 85 \times 37 = 1572.5 \text{ mm}^2$

Since there are four trapezoidal faces, the total lateral surface area is: $A_{\text{lateral}} = 4 \times 1572.5 = 6290 \text{ mm}^2$

3. Total Surface Area:

The total surface area of the frustum is the sum of the top, bottom, and lateral surface areas: $A_{\text{total}} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{lateral}} = 1369 + 2304 + 6290 = 9963 \text{ mm}^2$

Rounding to 1 decimal place, the surface area of the frustum is: $A_{\text{total}} \approx 9963.0 \text{ mm}^2$

Would you like more details or clarifications? Here are some additional questions for you:

1. What is the formula for the volume of a frustum?
2. How does the surface area of a frustum change if the slant height increases?
3. Can we calculate the volume of this frustum with the given data?
4. What is the relationship between a pyramid and its frustum?
5. How would the surface area formula differ for a frustum of a cone?

Tip: Always check the unit consistency when working with area and volume problems.