To find the surface area of a cone, we need to calculate both the lateral surface area and the area of the base. The formula for the total surface area $A$ of a cone is:

$A = \pi r (r + l)$

Where:

• $r$ is the radius of the base (in this case, 3 cm),
• $l$ is the slant height of the cone,
• $\pi$ is a constant (approximately 3.1416).

Step 1: Calculate the Slant Height $l$

The slant height can be calculated using the Pythagorean theorem since we are given the height and the radius of the base:

$l = \sqrt{r^2 + h^2}$

Given:

• $r = 3 \, \text{cm}$,
• $h = 9 \, \text{cm}$ (height of the cone).

Now, let's calculate the slant height.

$l = \sqrt{3^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} \approx 9.49 \, \text{cm}$

Step 2: Calculate the Surface Area

Now that we have the slant height, we can plug it into the formula for the surface area:

$A = \pi \times 3 \times (3 + 9.49) = \pi \times 3 \times 12.49$

Let's calculate the total surface area.

$A \approx 3.1416 \times 3 \times 12.49 \approx 117.76 \, \text{cm}^2$

Conclusion:

The surface area of the cone is approximately 117.76 cm².

Do you have any questions, or would you like a more detailed breakdown? Here are some related questions to deepen your understanding:

1. What is the formula for the volume of a cone?
2. How would the surface area change if the radius were doubled?
3. Can the surface area of a cone ever be smaller than the area of its base?
4. How does the slant height affect the surface area of a cone?
5. What is the difference between the lateral surface area and the total surface area of a cone?

Tip: Always make sure to calculate the slant height correctly when solving for the surface area of cones, as it combines both the height and radius in the Pythagorean theorem.