Question

A cone has the radius of six and the height of 11 what is the surface area of the cone?]

Answer 1

To find the surface area of a cone, you can use the formula:

\[
\text{Surface Area} = \pi r (r + l)
\]

where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.

First, we need to calculate the slant height \( l \) using the Pythagorean theorem. The slant height can be found with the formula:

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

where \( h \) is the height of the cone. Given:
- \( r = 6 \)
- \( h = 11 \)

Let's calculate the slant height \( l \):

\[
l = \sqrt{6^2 + 11^2} = \sqrt{36 + 121} = \sqrt{157}
\]

Next, we can calculate the surface area:

\[
\text{Surface Area} = \pi r (r + l) = \pi \times 6 \times (6 + \sqrt{157})
\]

Calculating \( 6 + \sqrt{157} \):

1. Calculate \( \sqrt{157} \) (approximately \( 12.53 \)):
\[
6 + \sqrt{157} \approx 6 + 12.53 \approx 18.53
\]

2. Now, substitute back into the surface area formula:
\[
\text{Surface Area} \approx \pi \times 6 \times 18.53
\]
\[
\text{Surface Area} \approx 6 \times 18.53 \approx 111.18
\]
\[
\text{Surface Area} \approx 111.18 \pi
\]

3. Finally, using \( \pi \approx 3.14 \):
\[
\text{Surface Area} \approx 111.18 \times 3.14 \approx 349.07
\]

Thus, the surface area of the cone is approximately:

\[
\text{Surface Area} \approx 349.07 \text{ square units}
\]

If you want the answer in terms of \( \pi \), it would be:

\[
\text{Surface Area} \approx 111.18 \pi \text{ square units}
\]