https://file.aiquickdraw.com/mathbot/file/1727370509599t37b359z.jpg

We are asked to find the surface area of the cone in the image, and we are to leave the answer in terms of \(\pi\).

The formula for the surface area (SA) of a cone is given as:
\[
SA = \pi r l + \pi r^2
\]
where:
- \(r\) is the radius of the base,
- \(l\) is the slant height, and
- \(\pi r l\) represents the lateral surface area,
- \(\pi r^2\) represents the area of the base.

### Step 1: Identify the values from the image.
From the image:
- The radius \(r = 12 \, \text{cm}\),
- The vertical height \(h = 16 \, \text{cm}\) is given, but the slant height \(l\) is unknown.

### Step 2: Calculate the slant height \(l\).
Using the Pythagorean theorem, we can find the slant height \(l\) because the height, radius, and slant height form a right triangle. The formula is:
\[
l = \sqrt{h^2 + r^2}
\]
Substituting the given values:
\[
l = \sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20 \, \text{cm}
\]

### Step 3: Substitute the values into the surface area formula.
Now that we have \(r = 12 \, \text{cm}\) and \(l = 20 \, \text{cm}\), we can substitute these into the surface area formula:
\[
SA = \pi (12)(20) + \pi (12)^2
\]
\[
SA = 240\pi + 144\pi
\]
\[
SA = 384\pi \, \text{cm}^2
\]

### Final Answer:
The surface area of the cone is:
\[
SA = 384\pi \, \text{cm}^2
\]

Would you like further details or have any questions?

Here are five related questions you might want to explore:
1. How does the surface area change if the radius is doubled but the height stays the same?
2. What is the lateral surface area of this cone?
3. How would you calculate the volume of this cone?
4. What is the ratio of the lateral surface area to the total surface area?
5. Can you derive the surface area formula from scratch using calculus?

**Tip:** For any cone, remember that the slant height is always the hypotenuse of the right triangle formed by the height and radius.

Find the surface area of the cone below. Leave your answer in terms of π. The cone has a radius of 12 cm and a height of 16 cm.

Learn how to calculate the surface area of a cone with a radius of 12 cm and a height of 16 cm. Using the Pythagorean theorem to find the slant height, this step-by-step guide walks you through applying the surface area formula in geometry. Suitable for Grades 9-11.

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