1. **Problem Statement:** The base of the lampshade is a regular hexagon with side length 8 inches. We need to estimate the amount of glass needed to make the lampshade, which means finding the surface area of the lampshade. 2. **Understanding the shape:** A lampshade with a regular hexagonal base is a hexagonal pyramid. To find the surface area, we need the area of the base plus the area of the 6 triangular faces. 3. **Formula for the area of a regular hexagon:** $$\text{Area}_{base} = \frac{3\sqrt{3}}{2} s^2$$ where $s$ is the side length. 4. **Calculate the base area:** $$\text{Area}_{base} = \frac{3\sqrt{3}}{2} \times 8^2 = \frac{3\sqrt{3}}{2} \times 64 = 96\sqrt{3} \approx 166.28 \text{ in}^2$$ 5. **Find the slant height:** The problem does not give the slant height directly, but the lamp on the right shows a height of 10 inches (assumed vertical height). We can find the slant height $l$ using the Pythagorean theorem. 6. **Calculate the apothem (inradius) of the hexagon:** $$a = s \times \frac{\sqrt{3}}{2} = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} \approx 6.93 \text{ in}$$ 7. **Calculate slant height $l$:** $$l = \sqrt{h^2 + a^2} = \sqrt{10^2 + (6.93)^2} = \sqrt{100 + 48.02} = \sqrt{148.02} \approx 12.17 \text{ in}$$ 8. **Calculate the lateral surface area:** The lateral surface area is the sum of the areas of 6 triangles, each with base $s=8$ in and height $l=12.17$ in. $$\text{Lateral area} = 6 \times \frac{1}{2} \times 8 \times 12.17 = 3 \times 8 \times 12.17 = 292.08 \text{ in}^2$$ 9. **Calculate total surface area:** $$\text{Total surface area} = \text{Area}_{base} + \text{Lateral area} = 166.28 + 292.08 = 458.36 \text{ in}^2$$ **Final answer:** The estimated amount of glass needed to make the lampshade is approximately **458.36 square inches**.