1. **State the problem:** We have a frustum formed by removing a smaller square-based pyramid from a similar larger one. The base edge of the larger pyramid is 48 mm, and the slant height is split into two equal parts of 37 mm each, so the total slant height of the original pyramid is $37 + 37 = 74$ mm. 2. **Identify what to find:** We need to find the surface area of the frustum. This includes the areas of the two square bases (top and bottom) and the four trapezium-shaped lateral faces. 3. **Key formulas and rules:** - Surface area of frustum = Area of bottom square + Area of top square + Area of 4 trapezium faces. - Since the pyramids are similar, the ratio of corresponding edges is the same. - The trapezium faces are all equal in area. 4. **Find the scale factor for the smaller pyramid:** - The total slant height of the original pyramid is 74 mm. - The smaller pyramid is removed by cutting at the midpoint of the slant height, so the smaller pyramid's slant height is 37 mm. - Scale factor $k = \frac{37}{74} = \frac{1}{2}$. 5. **Find the top square edge length:** - Since the pyramids are similar, the top square edge length is half the bottom edge length. - Top edge length $= 48 \times \frac{1}{2} = 24$ mm. 6. **Calculate areas of the bases:** - Bottom square area $= 48^2 = 2304$ mm$^2$. - Top square area $= 24^2 = 576$ mm$^2$. 7. **Calculate the height of the frustum:** - The height of the original pyramid $h$ can be found using Pythagoras in the right triangle formed by height, half base, and slant height. - Half base $= \frac{48}{2} = 24$ mm. - Slant height $= 74$ mm. - Height $h = \sqrt{74^2 - 24^2} = \sqrt{5476 - 576} = \sqrt{4900} = 70$ mm. - Height of smaller pyramid $h_s = 70 \times \frac{1}{2} = 35$ mm. - Height of frustum $= 70 - 35 = 35$ mm. 8. **Calculate the lateral surface area:** - Each trapezium face has parallel sides equal to the bottom edge and top edge of the pyramid face. - The base of each trapezium is the edge of the square base divided by 4 (since 4 faces), but since the trapezium is the face itself, the parallel sides are the bottom edge and top edge of the face. - The trapezium's parallel sides are the bottom edge of the base (48 mm) and the top edge (24 mm). - The height of the trapezium is the slant height of the frustum, which is 37 mm (the difference between the two slant heights). - Area of one trapezium face: $$\text{Area} = \frac{(a + b)}{2} \times h = \frac{(48 + 24)}{2} \times 37 = \frac{72}{2} \times 37 = 36 \times 37 = 1332 \text{ mm}^2$$ - Total lateral area $= 4 \times 1332 = 5328$ mm$^2$. 9. **Calculate total surface area of the frustum:** $$\text{Surface area} = \text{bottom area} + \text{top area} + \text{lateral area} = 2304 + 576 + 5328 = 8208 \text{ mm}^2$$ **Final answer:** The surface area of the frustum is **8208 mm$^2$**.