A cone has a height of 
9
9 cm and a radius of 
4
4 cm. First, find the slant height of the cone, then calculate its lateral surface area.

To find the slant height (\(l\)) of a cone, we can use the Pythagorean theorem:

\[
l = \sqrt{r^2 + h^2}
\]

where:
- \(r\) is the radius (4 cm),
- \(h\) is the height (9 cm).

Calculating the slant height:

\[
l = \sqrt{4^2 + 9^2} = \sqrt{16 + 81} = \sqrt{97} \approx 9.85 \text{ cm}
\]

Next, we can calculate the lateral surface area (\(A\)) of the cone using the formula:

\[
A = \pi r l
\]

Substituting the values we have:

\[
A = \pi \times 4 \times \sqrt{97} \approx \pi \times 4 \times 9.85 \approx 123.28 \text{ cm}^2
\]

### Summary:
- **Slant height**: approximately \(9.85\) cm
- **Lateral surface area**: approximately \(123.28\) cm²

Would you like more details or have any questions? Here are some related questions to consider:

1. How do you derive the formula for the lateral surface area of a cone?
2. What is the total surface area of the cone?
3. How would the lateral surface area change if the radius doubled?
4. What is the volume of the cone?
5. How does the height of the cone affect its slant height? 

**Tip**: Remember that the slant height is crucial for calculating the lateral surface area, and always check your calculations for accuracy!

A cone has a height of 9 cm and a radius of 4 cm. First, find the slant height of the cone, then calculate its lateral surface area.

Learn how to calculate the slant height and lateral surface area of a cone with a height of 9 cm and a radius of 4 cm. This guide covers the application of the Pythagorean theorem and formulas for lateral surface area, providing key insights for students in Grades 7-10.

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