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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.

Find the surface area of the following cone with a base radius of 3 cm and a height of 9 cm.

Learn how to find the surface area of a cone with a radius of 3 cm and a height of 9 cm using the Pythagorean theorem to calculate the slant height. This solution involves the surface area formula A = πr(r + l). Suitable for students in Grades 8-10.

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