Let's solve this step-by-step using the formula for the surface area of a cone:

Formula:

$A = \pi r^2 + \pi r s$ Where:

• $r$ is the radius of the base
• $s$ is the slant height of the cone

Values from the diagram:

• $r = 3$ inches (radius)
• $s = 8$ inches (slant height)

Step 1: Compute the base area

$\pi r^2 = \pi (3)^2 = 9\pi$

Step 2: Compute the lateral surface area

$\pi r s = \pi (3)(8) = 24\pi$

Step 3: Total surface area

$A = \pi r^2 + \pi r s = 9\pi + 24\pi = 33\pi \, \text{square inches}$

Final Answer:

The minimum amount of construction paper needed is: $\boxed{33\pi \, \text{square inches}}$

Would you like a deeper explanation or assistance with related concepts?

Follow-up Questions:

1. How is the slant height of a cone calculated if it’s not given?
2. Can you explain the difference between lateral surface area and total surface area?
3. What would happen to the surface area if the radius is doubled?
4. How can this formula be adapted for a cone without a base?
5. Can you provide an example where the surface area is required in a real-world scenario?

Tip:

When solving geometry problems involving cones, always ensure you know whether the surface area includes the base or only the lateral area!