To find the surface area of a cone, we can use the formula:
\[ \text{Surface Area} = \pi r^2 + \pi r l \]
where:
- \( r \) = radius of the base of the cone,
- \( h \) = height of the cone,
- \( l \) = slant height of the cone.
First, we need to calculate the slant height \( l \) using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the given values \( r = 6 \) and \( h = 11 \):
\[ l = \sqrt{6^2 + 11^2} = \sqrt{36 + 121} = \sqrt{157} \]
Now, calculating \( \sqrt{157} \):
\[ \sqrt{157} \approx 12.53 \]
Now that we have \( l \), we can calculate the surface area:
\[ \text{Surface Area} = \pi r^2 + \pi r l \]
Substituting the values:
\[ \text{Surface Area} = \pi (6^2) + \pi (6)(12.53) \]
Calculating \( \pi (6^2) \):
\[ \pi (6^2) = \pi (36) \approx 3.14 \times 36 \approx 113.04 \]
Now calculating \( \pi (6)(12.53) \):
\[ \pi (6)(12.53) \approx 3.14 \times 6 \times 12.53 \approx 3.14 \times 75.18 \approx 236.60 \]
Now, adding both parts together:
\[ \text{Surface Area} \approx 113.04 + 236.60 \approx 349.64 \]
Finally, rounding to the nearest tenth:
\[ \text{Surface Area} \approx 349.6 \]
Thus, the surface area of the cone is approximately 349.6 square units.